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			<title>Using the Symbolics Menu</title>
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							<c val="#808000">GETTING STARTED PRIMERS</c>
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								<link href=".\..\..\toc_tuto.xmcd" title="Links to Tutorials" popup="false">tutorials</link>
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								<c val="#800000">Using Symbolic Calculation</c>
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				<phrase>Symbolics menu</phrase>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When you evaluate an expression numerically using =, Mathcad returns one or more numbers.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When you evaluate an expression symbolically, on the other hand, Mathcad returns the result as another expression, in terms of the variables and symbols in the original problem.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are two ways to evaluate an expression symbolically:</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can use the <b>
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							<link href="#symbolicequals" title="Links to section" popup="false">symbolic equal sign</link>
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					</b> (<f family="Symbol" charset="2">®</f>) as above. This is the recommended method because the results are "live" and update along with any changes you make in the worksheet.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can use commands from the <b>
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							<link href="#symbolicsmenu" title="Links to section" popup="false">Symbolics menu</link>
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					</b>. These results are "static" and are not affected by other definitions or changes in the worksheet. </p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This tutorial explains how to perform symbolic operations on expressions and evaluate the results.</p>
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				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using the Symbolic Equal Sign</p>
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			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To insert the symbolic equal sign after an expression, click anywhere in the expression and either:</p>
			</text>
		</region>
		<region region-id="2445" left="18" top="588" width="168" height="72.75" align-x="18" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<f size="12">
						<b>
							<c val="#008080">•</c>
						</b>
					</f>
					<f size="14">
						<b>
							<c val="#008080">
								<sp count="2"/>
							</c>
						</b>
					</f>Press <b>[Ctrl] [.]</b>.</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">- or -</p>
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="14.25" margin-right="0" text-indent="-14.25" text-align="left" list-style-type="none" tabs="13.5">
					<f size="12">
						<b>
							<c val="#008080">•</c>
						</b>
					</f>
					<f size="14">
						<b>
							<c val="#008080">
								<sp count="2"/>
							</c>
						</b>
					</f>Click <f family="Symbol" charset="2">® </f>
					<f family="Times New Roman" charset="0">on the </f>
					<f family="Times New Roman" charset="0">
						<b>Symbolic</b>
					</f>
					<f family="Times New Roman" charset="0"> toolbar.</f>
				</p>
			</text>
		</region>
		<region region-id="2619" left="6" top="662.25" width="357" height="37.5" align-x="6" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<f family="Times New Roman" charset="0" size="12">
						<c val="#000">For example, to insert the symbolic equal sign after the expression</c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">
							<sp/>
						</c>
					</f>
					<region region-id="2618" left="305.25" top="663" width="41.25" height="36.75" align-x="305.25" align-y="684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:derivative/>
								<ml:lambda>
									<ml:boundVars>
										<ml:id xml:space="preserve">x</ml:id>
									</ml:boundVars>
									<ml:apply>
										<ml:id xml:space="preserve">sin</ml:id>
										<ml:id xml:space="preserve">x</ml:id>
									</ml:apply>
								</ml:lambda>
							</ml:apply>
						</math>
						<rendering item-idref="7"/>
					</region>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">:</c>
						</inlineAttr>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2473" left="6" top="704.25" width="348" height="64.5" align-x="6" align-y="714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal">
							<c val="#800080">1. Click the expression.</c>
						</inlineAttr>
					</f>
				</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal">
							<c val="#800080">2. Press </c>
						</inlineAttr>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">
							<b>[Ctrl] [.]</b>
						</c>
					</f>
					<inlineAttr font-weight="normal">
						<c val="#800080">
							<f family="Tahoma" charset="0" size="10"> to insert the symbolic equal sign </f>
						</c>
					</inlineAttr>
					<f family="Symbol" charset="2" size="10">
						<c val="#800080">®</c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal">
							<c val="#800080">. </c>
						</inlineAttr>
					</f>
				</p>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit"/>
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal">
							<c val="#800080">3. Press </c>
						</inlineAttr>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<b>
							<c val="#800080">[Enter]</c>
						</b>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal">
							<c val="#800080"> to evaluate the expression. </c>
						</inlineAttr>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2520" left="6" top="776.25" width="20.25" height="12" align-x="6" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2630" left="18" top="801" width="90" height="36.75" align-x="66.75" align-y="822" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:derivative/>
						<ml:lambda>
							<ml:boundVars>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:boundVars>
							<ml:apply>
								<ml:id xml:space="preserve">sin</ml:id>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
						</ml:lambda>
					</ml:apply>
					<ml:symResult>
						<ml:apply>
							<ml:id xml:space="preserve">cos</ml:id>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="8"/>
		</region>
		<region region-id="2526" left="6" top="846.75" width="344.25" height="42.75" align-x="6" align-y="858" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Times New Roman" charset="0" size="12">
						<c val="#000">Mathcad displays the result to the right of the symbolic equal sign. T</c>
					</f>he result is "live." If you make changes to the expression, the result is automatically updated. See <b>
						<u>
							<link href="#livesymbolics" title="Links to section" popup="false">Live Symbolics</link>
						</u>
					</b>.</p>
			</text>
		</region>
		<region region-id="2488" left="6" top="900.75" width="341.25" height="28.5" align-x="6" align-y="912" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Another way to insert the symbolic equal sign is to select <b>Toolbars &gt; </b>
					<b>Symbolic</b> from the <b>View</b> menu to open the <b>Symbolic</b> toolbar, </p>
			</text>
		</region>
		<region region-id="2695" left="6" top="942" width="353.25" height="110.25" align-x="6" align-y="942" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<picture>
				<png item-idref="9" display-width="351.75" display-height="108.75"/>
			</picture>
			<rendering item-idref="10"/>
		</region>
		<region region-id="2462" left="0" top="1062.75" width="3" height="14.25" align-x="10.5" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<sp/>
				</p>
			</text>
		</region>
		<region region-id="2483" left="6" top="1062" width="303" height="15" align-x="6" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">and then click the right arrow <f family="Symbol" charset="2">® </f>on the toolbar.</p>
			</text>
		</region>
		<region region-id="2620" left="0" top="1092" width="6000" height="6" align-x="0" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="2497" left="6" top="1100.25" width="237.75" height="12" align-x="24.75" align-y="1110" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="symbolicsmenu">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using the Symbolics Menu</p>
			</text>
		</region>
		<region region-id="2484" left="6" top="1122.75" width="341.25" height="57" align-x="6" align-y="1134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can also evaluate an expression symbolically by selecting it and then selecting <b>Evaluate &gt; Symbolically</b> from the <b>Symbolics</b> menu. However, in this case the results are not automatically updated when you make changes elsewhere in the worksheet.</p>
			</text>
		</region>
		<region region-id="2587" left="6" top="1196.25" width="191.25" height="12" align-x="6" align-y="1206" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="exactevaluation">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Evaluating an Expression Exactly</p>
			</text>
			<indexes>
				<phrase>exact evaluations</phrase>
			</indexes>
		</region>
		<region region-id="2678" left="6" top="1218.75" width="390.75" height="61.5" align-x="26.25" align-y="1230" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The numerical equal sign (=) returns answers as decimal numbers, approximating them to a fixed number of decimal places when necessary. The symbolic equal sign, on the other hand, returns exact results, using ordinary fractions and symbolic representations of irrational numbers, such as <region region-id="2716" left="140.25" top="1262.25" width="15.75" height="18" align-x="142.5" align-y="1275.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:sqrt/>
								<ml:real>2</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="11"/>
					</region> or π.</p>
			</text>
		</region>
		<region region-id="2564" left="6" top="1304.25" width="185.25" height="36" align-x="17.25" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Click the expression below and then click <f family="Symbol" charset="2">®</f> on the <b>Symbolic</b> toolbar. Press <b>[Enter]</b> to see the result:</p>
			</text>
		</region>
		<region region-id="2565" left="228" top="1302.75" width="146.25" height="115.5" align-x="234.75" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">Tip</c>
					</b>: The symbolic equal sign (<f family="Symbol" charset="2">®</f>) gives an exact result for any numeric operations you want to perform, including simplification of expressions involving fractions, roots, <f family="Symbol" charset="2">p</f>, logarithms, exponents, and trigonometric functions.</p>
			</text>
			<indexes>
				<phrase>symbolic result sign</phrase>
			</indexes>
		</region>
		<region region-id="2640" left="18" top="1354.5" width="1.169422E+07" height="33.75" align-x="121.5" align-y="1374" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:plus/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:plus/>
									<ml:apply>
										<ml:div/>
										<ml:real>2</ml:real>
										<ml:real>3</ml:real>
									</ml:apply>
									<ml:apply>
										<ml:div/>
										<ml:real>3</ml:real>
										<ml:real>4</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:real>4</ml:real>
									<ml:real>5</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>5</ml:real>
								<ml:real>6</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:div/>
							<ml:real>6</ml:real>
							<ml:real>7</ml:real>
						</ml:apply>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:apply>
							<ml:div/>
							<ml:real>7</ml:real>
							<ml:real>8</ml:real>
						</ml:apply>
						<ml:id xml:space="preserve">π</ml:id>
					</ml:apply>
				</ml:apply>
			</math>
		</region>
		<region region-id="2567" left="6" top="1424.25" width="20.25" height="12" align-x="13.5" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2643" left="18" top="1444.5" width="2.338524E+07" height="33.75" align-x="1.16927E+07" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:plus/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:div/>
											<ml:real>2</ml:real>
											<ml:real>3</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:div/>
											<ml:real>3</ml:real>
											<ml:real>4</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:div/>
										<ml:real>4</ml:real>
										<ml:real>5</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:real>5</ml:real>
									<ml:real>6</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>6</ml:real>
								<ml:real>7</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:div/>
								<ml:real>7</ml:real>
								<ml:real>8</ml:real>
							</ml:apply>
							<ml:id xml:space="preserve">π</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:symResult>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:mult/>
									<ml:real>7</ml:real>
									<ml:id xml:space="preserve">π</ml:id>
								</ml:apply>
								<ml:real>8</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>83</ml:real>
								<ml:real>140</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
		</region>
		<region region-id="2583" left="6" top="1494.75" width="343.5" height="28.5" align-x="17.25" align-y="1506" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note the difference between this and the result returned by the numerical equal sign:</p>
			</text>
		</region>
		<region region-id="2631" left="18" top="1534.5" width="1.169272E+07" height="33.75" align-x="1.16927E+07" align-y="1554" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:plus/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:div/>
											<ml:real>2</ml:real>
											<ml:real>3</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:div/>
											<ml:real>3</ml:real>
											<ml:real>4</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:div/>
										<ml:real>4</ml:real>
										<ml:real>5</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:real>5</ml:real>
									<ml:real>6</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>6</ml:real>
								<ml:real>7</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:div/>
								<ml:real>7</ml:real>
								<ml:real>8</ml:real>
							</ml:apply>
							<ml:id xml:space="preserve">π</ml:id>
						</ml:apply>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:real>3.3417507147482115</ml:real>
					</result>
				</ml:eval>
			</math>
		</region>
		<region region-id="2613" left="0" top="1578" width="6000" height="6" align-x="0" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="2590" left="6" top="1586.25" width="191.25" height="12" align-x="6" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="livesymbolics">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Live Symbolics</p>
			</text>
		</region>
		<region region-id="2591" left="6" top="1608.75" width="347.25" height="28.5" align-x="6" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">The symbolic equal sign returns "live" results. If you change an expression, or make changes above or to the left of it, the result is automatically updated. </p>
			</text>
		</region>
		<region region-id="2569" left="6" top="1652.25" width="264.75" height="24" align-x="8.25" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Change one of the numbers in the equation below and watch the result change. </p>
			</text>
		</region>
		<region region-id="2642" left="18" top="1690.5" width="2.338524E+07" height="33.75" align-x="1.16927E+07" align-y="1710" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:plus/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:div/>
											<ml:real>2</ml:real>
											<ml:real>3</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:div/>
											<ml:real>3</ml:real>
											<ml:real>4</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:div/>
										<ml:real>4</ml:real>
										<ml:real>5</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:div/>
									<ml:real>5</ml:real>
									<ml:real>6</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>6</ml:real>
								<ml:real>7</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:div/>
								<ml:real>7</ml:real>
								<ml:real>8</ml:real>
							</ml:apply>
							<ml:id xml:space="preserve">π</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:symResult>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:mult/>
									<ml:real>7</ml:real>
									<ml:id xml:space="preserve">π</ml:id>
								</ml:apply>
								<ml:real>8</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:real>83</ml:real>
								<ml:real>140</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
		</region>
		<region region-id="2570" left="6" top="1734" width="285" height="29.25" align-x="12.75" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The symbolic equal sign (<f family="Symbol" charset="2">®</f>) uses any variables or functions defined in your worksheet when calculating results. </p>
			</text>
		</region>
		<region region-id="2627" left="18" top="1771.5" width="48.75" height="21" align-x="45.75" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:function>
						<ml:id xml:space="preserve">f</ml:id>
						<ml:boundVars>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:boundVars>
					</ml:function>
					<ml:apply>
						<ml:pow/>
						<ml:id xml:space="preserve">x</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="12"/>
		</region>
		<region region-id="2639" left="114" top="1776.75" width="39" height="15.75" align-x="127.5" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:real>2.5</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="13"/>
		</region>
		<region region-id="2634" left="18" top="1800.75" width="59.25" height="15.75" align-x="45.75" align-y="1812" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">f</ml:id>
						<ml:id xml:space="preserve">y</ml:id>
					</ml:apply>
					<ml:symResult>
						<ml:real>6.25</ml:real>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="14"/>
		</region>
		<region region-id="2500" left="6" top="1832.25" width="237.75" height="12" align-x="6" align-y="1842" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="Keywords">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using Keywords</p>
			</text>
		</region>
		<region region-id="2502" left="6" top="1854.75" width="334.5" height="42.75" align-x="12" align-y="1866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To perform specific symbolic operations, you can add <i>keywords</i> before the symbolic sign. The <b>Symbolic</b> toolbar contains a list of the available keywords.</p>
			</text>
		</region>
		<region region-id="2446" left="6" top="1902.75" width="277.5" height="21.75" align-x="16.5" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal">
							<c val="#000">For example, to solve the equation </c>
						</inlineAttr>
					</f>
					<region region-id="2503" left="168" top="1903.5" width="75.75" height="21" align-x="228.75" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:equal/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:pow/>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:mult style="auto-select"/>
											<ml:real>2</ml:real>
											<ml:id xml:space="preserve">x</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:real>3</ml:real>
								</ml:apply>
								<ml:real>0</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="15"/>
					</region>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">:</c>
						</inlineAttr>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal">
							<c val="#000">
								<sp/>
							</c>
						</inlineAttr>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2679" left="6" top="1934.25" width="36" height="12" align-x="6" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">1. Type:</c>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2659" left="24" top="1957.5" width="54" height="21" align-x="64.5" align-y="1974" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:minus/>
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:pow/>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:real>2</ml:real>
						</ml:apply>
						<ml:apply>
							<ml:mult style="auto-select"/>
							<ml:real>2</ml:real>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:real>3</ml:real>
				</ml:apply>
			</math>
			<rendering item-idref="16"/>
		</region>
		<region region-id="2660" left="18" top="1988.25" width="289.5" height="24" align-x="18" align-y="1998" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<b>
							<c val="#800080">Note:</c>
						</b>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080"> You only need to type the left side of the equation because the right side is 0.</c>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2663" left="6" top="2024.25" width="315" height="76.5" align-x="6" align-y="2034" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="none" tabs="13.5">
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">2.<sp count="2"/>Press </c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<b>
							<c val="#800080">[Ctrl] [Shift] [.]</c>
						</b>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080"> to insert a placeholder followed by the symbolic equal sig</c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">n. </c>
					</f>
				</p>
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="none" tabs="13.5"/>
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="none" tabs="13.5">
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">3. In the placeholder, type the keyword "solve."</c>
					</f>
				</p>
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="none" tabs="13.5"/>
				<p style="Normal" margin-left="13.5" margin-right="0" text-indent="-13.5" text-align="left" list-style-type="none" tabs="13.5">
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">4. Press </c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<b>
							<c val="#800080">[Enter]</c>
						</b>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">.</c>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2521" left="6" top="2108.25" width="20.25" height="12" align-x="6" align-y="2118" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2626" left="18" top="2126.25" width="132.75" height="36.75" align-x="111" align-y="2148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:pow/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>2</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:real>3</ml:real>
					</ml:apply>
					<ml:command>
						<ml:id xml:space="preserve">solve</ml:id>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:real>-1</ml:real>
							<ml:real>3</ml:real>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="17"/>
		</region>
		<region region-id="2523" left="6" top="2172.75" width="206.25" height="14.25" align-x="6" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Mathcad returns the two solutions in a vector.</p>
			</text>
		</region>
		<region region-id="2614" left="0" top="2190" width="6000" height="6" align-x="0" align-y="2190" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="1986" left="6" top="2202.75" width="348" height="50.25" align-x="14.25" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The expression or equation appears on the left, followed by the <inlineAttr font-weight="normal">keyword</inlineAttr> that describes the operation. In this case, the operation is to solve the equation <region region-id="1971" left="6" top="2227.5" width="75.75" height="21" align-x="66.75" align-y="2244" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:equal style="equal"/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:pow/>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:mult style="auto-select"/>
											<ml:real>2</ml:real>
											<ml:id xml:space="preserve">x</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:real>3</ml:real>
								</ml:apply>
								<ml:real>0</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="18"/>
					</region>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.</inlineAttr>
					</f> The solution follows the symbolic equal sign (<f family="Symbol" charset="2">®</f>).</p>
			</text>
		</region>
		<region region-id="2594" left="6" top="2264.25" width="162.75" height="12" align-x="24" align-y="2274" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Simplifying an Expression</p>
			</text>
		</region>
		<region region-id="2515" left="6" top="2286.75" width="342.75" height="36" align-x="6" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To simplify an expression, use the keyword "simplify."<f family="Times New Roman" charset="0" size="12">
						<sp/>
					</f>
					<f family="Times New Roman" charset="0" size="12">
						<c val="#000">For example, to simplify </c>
					</f>
					<region region-id="2516" left="42" top="2293.5" width="112.5" height="21" align-x="141" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:minus/>
								<ml:apply>
									<ml:plus/>
									<ml:apply>
										<ml:plus/>
										<ml:apply>
											<ml:pow/>
											<ml:parens>
												<ml:apply>
													<ml:minus/>
													<ml:id xml:space="preserve">x</ml:id>
													<ml:real>1</ml:real>
												</ml:apply>
											</ml:parens>
											<ml:real>2</ml:real>
										</ml:apply>
										<ml:apply>
											<ml:mult style="auto-select"/>
											<ml:real>2</ml:real>
											<ml:apply>
												<ml:pow/>
												<ml:id xml:space="preserve">x</ml:id>
												<ml:real>2</ml:real>
											</ml:apply>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:mult style="auto-select"/>
										<ml:real>4</ml:real>
										<ml:id xml:space="preserve">x</ml:id>
									</ml:apply>
								</ml:apply>
								<ml:real>1</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="19"/>
					</region>
					<f family="Times New Roman" charset="0" size="12">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#000">:</c>
						</inlineAttr>
					</f>
				</p>
			</text>
			<indexes>
				<phrase>simplify keyword</phrase>
				<phrase>expand keyword</phrase>
				<phrase>factor keyword</phrase>
			</indexes>
		</region>
		<region region-id="2519" left="6" top="2336.25" width="180.75" height="60" align-x="6" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1. Type the expression.</p>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">2. Click "simplify" on the Symbolic toolbar.</p>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3.<f family="Tahoma" charset="0" size="10">
						<c val="#800080"> Press </c>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<b>
							<c val="#800080">[Enter]</c>
						</b>
					</f>
					<f family="Tahoma" charset="0" size="10">
						<c val="#800080">.</c>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2595" left="6" top="2408.25" width="20.25" height="12" align-x="6" align-y="2418" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2647" left="24" top="2431.5" width="217.5" height="21" align-x="186" align-y="2448" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:minus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:plus/>
								<ml:apply>
									<ml:pow/>
									<ml:parens>
										<ml:apply>
											<ml:minus/>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:real>1</ml:real>
										</ml:apply>
									</ml:parens>
									<ml:real>2</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:mult style="auto-select"/>
									<ml:real>2</ml:real>
									<ml:apply>
										<ml:pow/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult style="auto-select"/>
								<ml:real>4</ml:real>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:real>1</ml:real>
					</ml:apply>
					<ml:command>
						<ml:id xml:space="preserve">simplify</ml:id>
					</ml:command>
					<ml:symResult>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">x</ml:id>
							<ml:parens>
								<ml:apply>
									<ml:plus/>
									<ml:apply>
										<ml:mult/>
										<ml:real>3</ml:real>
										<ml:id xml:space="preserve">x</ml:id>
									</ml:apply>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:parens>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="20"/>
		</region>
		<region region-id="1880" left="6" top="2468.25" width="331.5" height="45.75" align-x="17.25" align-y="2478" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Alternatively, put the blue insertion line at the end of the expression and then click <f size="18">
						<b>.</b>
					</f>
					<f family="Symbol" charset="2">®</f> on the <b>Symbolic</b> toolbar. In the empty placeholder, type the keyword "<inlineAttr font-weight="normal">simplify."</inlineAttr> Press <b>[Enter]</b> to see the result.</p>
			</text>
		</region>
		<region region-id="2615" left="0" top="2532" width="6000" height="6" align-x="0" align-y="2532" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="2312" left="6" top="2534.25" width="283.5" height="12" align-x="6" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Viewing Approximate and Intermediate Expressions</p>
			</text>
		</region>
		<region region-id="2340" left="6" top="2556.75" width="342" height="28.5" align-x="6" align-y="2568" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To see the values of numerically defined variables to a certain precision, use the keyword "float," as shown in the following example. </p>
			</text>
		</region>
		<region region-id="2706" left="6" top="2600.25" width="233.25" height="12" align-x="6" align-y="2610" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1. Define y by typing</p>
			</text>
		</region>
		<region region-id="2704" left="18" top="2622.75" width="39" height="15.75" align-x="31.5" align-y="2634" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">y</ml:id>
					<ml:real>2.5</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="21"/>
		</region>
		<region region-id="2339" left="6" top="2654.25" width="233.25" height="24" align-x="17.25" align-y="2664" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">2. Click the expression below and then click <f family="Symbol" charset="2">®</f> on the <b>Symbolic</b> toolbar. Press <b>[Enter]</b> to see the result.</p>
			</text>
		</region>
		<region region-id="2625" left="18" top="2692.5" width="28.5" height="33.75" align-x="33" align-y="2712" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:plus/>
					<ml:apply>
						<ml:div/>
						<ml:id xml:space="preserve">y</ml:id>
						<ml:real>3</ml:real>
					</ml:apply>
					<ml:id xml:space="preserve">a</ml:id>
				</ml:apply>
			</math>
			<rendering item-idref="22"/>
		</region>
		<region region-id="2341" left="6" top="2738.25" width="20.25" height="12" align-x="13.5" align-y="2748" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2629" left="18" top="2758.5" width="194.25" height="33.75" align-x="54" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">y</ml:id>
							<ml:real>3</ml:real>
						</ml:apply>
						<ml:id xml:space="preserve">a</ml:id>
					</ml:apply>
					<ml:symResult>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">a</ml:id>
							<ml:real>0.83333333333333333333</ml:real>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="23"/>
		</region>
		<region region-id="2357" left="6" top="2808.75" width="251.25" height="28.5" align-x="12.75" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">Tip</c>
					</b>: By default, values with decimals trigger symbolic floating point calculation to 20 digits.</p>
			</text>
			<indexes>
				<phrase>symbolic result sign</phrase>
			</indexes>
		</region>
		<region region-id="2347" left="6" top="2852.25" width="267" height="48" align-x="17.25" align-y="2862" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3. Click the expression above, place the blue insertion line between the <f family="Symbol" charset="2">®</f> and the a, and click <b>float</b> on the <b>Symbolic</b> toolbar. Type the number 4 in the empty placeholder. Press <b>[Enter]</b> to see the result.</p>
			</text>
		</region>
		<region region-id="2350" left="6" top="2912.25" width="20.25" height="12" align-x="13.5" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2637" left="18" top="2932.5" width="138" height="33.75" align-x="93.75" align-y="2952" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">y</ml:id>
							<ml:real>3</ml:real>
						</ml:apply>
						<ml:id xml:space="preserve">a</ml:id>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">float</ml:id>
							<ml:real>4</ml:real>
						</ml:sequence>
					</ml:command>
					<ml:symResult>
						<ml:apply>
							<ml:plus/>
							<ml:id xml:space="preserve">a</ml:id>
							<ml:real>0.8333</ml:real>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="24"/>
		</region>
		<region region-id="2700" left="6" top="2978.25" width="164.25" height="12" align-x="6" align-y="2988" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Mathcad displays the result to 4 digits.</p>
			</text>
		</region>
		<region region-id="2709" left="6" top="3000.75" width="341.25" height="42.75" align-x="6" align-y="3012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now, suppose you want to display the above expression, including the numerical value of y, but without performing any other arithmetic operations on the expression. To do so, </p>
			</text>
		</region>
		<region region-id="2358" left="6" top="3050.25" width="237.75" height="36" align-x="17.25" align-y="3060" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Delete "<inlineAttr font-weight="normal">float, 4"</inlineAttr> in the expression above and replace it with the keyword "<inlineAttr font-weight="normal">explicit,"</inlineAttr> followed by comma an the variable y. Press <b>[Enter]</b> to see the result.</p>
			</text>
		</region>
		<region region-id="2377" left="6" top="3092.25" width="20.25" height="12" align-x="13.5" align-y="3102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2650" left="18" top="3112.5" width="134.25" height="33.75" align-x="105" align-y="3132" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">y</ml:id>
							<ml:real>3</ml:real>
						</ml:apply>
						<ml:id xml:space="preserve">a</ml:id>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">explicit</ml:id>
							<ml:id xml:space="preserve">y</ml:id>
						</ml:sequence>
					</ml:command>
					<ml:symResult>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:div/>
								<ml:real>2.5</ml:real>
								<ml:real>3</ml:real>
							</ml:apply>
							<ml:id xml:space="preserve">a</ml:id>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="25"/>
		</region>
		<region region-id="2714" left="0" top="3150" width="6000" height="6" align-x="0" align-y="3150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="2713" left="6" top="3156.75" width="375" height="14.25" align-x="25.5" align-y="3168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad replaces y with 2.5, but does not perform any other arithmetic operations.</p>
			</text>
		</region>
		<region region-id="2272" left="6" top="3182.25" width="131.25" height="12" align-x="6" align-y="3192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Factoring an Expression</p>
			</text>
		</region>
		<region region-id="2601" left="6" top="3204.75" width="290.25" height="14.25" align-x="12" align-y="3216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To factor an expression, use the keyword "factor." For example,</p>
			</text>
		</region>
		<region region-id="1317" left="6" top="3222.75" width="212.25" height="103.5" align-x="17.25" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1. Type <region region-id="2602" left="42" top="3223.5" width="90" height="21" align-x="117.75" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
						<math optimize="false" disable-calc="false">
							<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
								<ml:plus/>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:mult/>
											<ml:real>2</ml:real>
											<ml:apply>
												<ml:pow/>
												<ml:id xml:space="preserve">x</ml:id>
												<ml:real>3</ml:real>
											</ml:apply>
										</ml:apply>
										<ml:apply>
											<ml:mult/>
											<ml:real>2</ml:real>
											<ml:id xml:space="preserve">x</ml:id>
										</ml:apply>
									</ml:apply>
									<ml:apply>
										<ml:mult/>
										<ml:real>3</ml:real>
										<ml:apply>
											<ml:pow/>
											<ml:id xml:space="preserve">x</ml:id>
											<ml:real>2</ml:real>
										</ml:apply>
									</ml:apply>
								</ml:apply>
								<ml:real>3</ml:real>
							</ml:apply>
						</math>
						<rendering item-idref="26"/>
					</region>
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#800080">.</c>
						</inlineAttr>
					</f>
				</p>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<f family="Tahoma" charset="0" size="10">
						<inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">
							<c val="#800080">2. C</c>
						</inlineAttr>
					</f>lick <f size="18">
						<b>.</b>
					</f>
					<f family="Symbol" charset="2">®</f> on the <b>Symbolic</b> toolbar. </p>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">3. In the empty placeholder, type the keyword "factor."</p>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">4. Press <b>[Enter]</b> to see the result. </p>
			</text>
		</region>
		<region region-id="2605" left="6" top="3350.25" width="20.25" height="12" align-x="13.5" align-y="3360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2633" left="18" top="3373.5" width="248.25" height="21" align-x="150" align-y="3390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:minus/>
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:mult/>
									<ml:real>2</ml:real>
									<ml:apply>
										<ml:pow/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>3</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:mult/>
									<ml:real>2</ml:real>
									<ml:id xml:space="preserve">x</ml:id>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:real>3</ml:real>
								<ml:apply>
									<ml:pow/>
									<ml:id xml:space="preserve">x</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:real>3</ml:real>
					</ml:apply>
					<ml:command>
						<ml:id xml:space="preserve">factor</ml:id>
					</ml:command>
					<ml:symResult>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:mult/>
								<ml:parens>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:mult/>
											<ml:real>2</ml:real>
											<ml:id xml:space="preserve">x</ml:id>
										</ml:apply>
										<ml:real>3</ml:real>
									</ml:apply>
								</ml:parens>
								<ml:parens>
									<ml:apply>
										<ml:minus/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>1</ml:real>
									</ml:apply>
								</ml:parens>
							</ml:apply>
							<ml:parens>
								<ml:apply>
									<ml:plus/>
									<ml:id xml:space="preserve">x</ml:id>
									<ml:real>1</ml:real>
								</ml:apply>
							</ml:parens>
						</ml:apply>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="27"/>
		</region>
		<region region-id="2276" left="6" top="3416.25" width="264" height="12" align-x="6" align-y="3426" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Solving an Equation in Two Variables</p>
			</text>
			<indexes>
				<phrase>equations, solving with symbolics menu</phrase>
				<phrase>solving equations, symbolics menu</phrase>
				<phrase>variables, solving symbolically</phrase>
			</indexes>
		</region>
		<region region-id="2278" left="6" top="3432.75" width="337.5" height="42.75" align-x="6" align-y="3444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can use symbolics to find closed-form solutions to a variety of problems, which may yield insight into the form and behavior of the solution. It is also possible to plot these closed-form solutions.</p>
			</text>
		</region>
		<region region-id="2273" left="6" top="3494.25" width="187.5" height="12" align-x="17.25" align-y="3504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1. Type:</p>
			</text>
		</region>
		<region region-id="2621" left="18" top="3519" width="123" height="10.5" align-x="18" align-y="3528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<f size="9">
						<b>
							<c val="#000"> z-3 [Ctrl]= x+2 [Shift] ^2</c>
						</b>
					</f>
				</p>
			</text>
		</region>
		<region region-id="2606" left="6" top="3536.25" width="20.25" height="12" align-x="6" align-y="3546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2636" left="18" top="3553.5" width="78.75" height="21" align-x="48.75" align-y="3570" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:equal/>
					<ml:apply>
						<ml:minus/>
						<ml:id xml:space="preserve">z</ml:id>
						<ml:real>3</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:pow/>
						<ml:parens>
							<ml:apply>
								<ml:plus/>
								<ml:id xml:space="preserve">x</ml:id>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:parens>
						<ml:real>2</ml:real>
					</ml:apply>
				</ml:apply>
			</math>
			<rendering item-idref="28"/>
		</region>
		<region region-id="2623" left="18" top="3590.25" width="288.75" height="12" align-x="18" align-y="3600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">
					<b>Note:</b>
					<inlineAttr font-weight="normal"> Use the Boolean equals (</inlineAttr>
					<b>[Ctrl] [</b>
					<b>=]</b>
					<inlineAttr font-weight="normal">) to specify the equation</inlineAttr>.</p>
			</text>
		</region>
		<region region-id="2275" left="6" top="3614.25" width="212.25" height="72" align-x="16.5" align-y="3624" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">2. Click <b>solve</b> on the <b>Symbolic</b> toolbar<inlineAttr font-weight="normal">.</inlineAttr>
				</p>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
					<inlineAttr font-weight="normal">3. Type a comma after </inlineAttr>
					<b>solve</b>
					<inlineAttr font-weight="normal">.</inlineAttr>
				</p>
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/>
				<p style="directions" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">4<inlineAttr font-weight="normal">. Type</inlineAttr> x in the placeholder to specify the variable you want to solve for. </p>
			</text>
		</region>
		<region region-id="2653" left="6" top="3698.25" width="75.75" height="12" align-x="6" align-y="3708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="false">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">5. Press <b>[Enter].</b>
				</p>
			</text>
		</region>
		<region region-id="2379" left="18" top="3720.75" width="27.75" height="14.25" align-x="18" align-y="3732" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
			</text>
		</region>
		<region region-id="2284" left="0" top="3744" width="6000" height="6" align-x="0" align-y="3744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<pageBreak/>
		</region>
		<region region-id="2380" left="6" top="3752.25" width="20.25" height="12" align-x="13.5" align-y="3762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2287" left="258" top="3780.75" width="141.75" height="71.25" align-x="258" align-y="3792" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
					<b>
						<c val="#008080">Tip</c>
					</b>: When the equation you are solving has more than one variable, specifiy the variable you want to solve for after the keyword "solve." </p>
			</text>
		</region>
		<region region-id="2646" left="18" top="3786" width="213.75" height="41.25" align-x="147" align-y="3810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:minus/>
							<ml:id xml:space="preserve">z</ml:id>
							<ml:real>3</ml:real>
						</ml:apply>
						<ml:apply>
							<ml:pow/>
							<ml:parens>
								<ml:apply>
									<ml:plus/>
									<ml:id xml:space="preserve">x</ml:id>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:parens>
							<ml:real>2</ml:real>
						</ml:apply>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">x</ml:id>
						</ml:sequence>
					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:sqrt/>
									<ml:apply>
										<ml:minus/>
										<ml:id xml:space="preserve">z</ml:id>
										<ml:real>3</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:real>2</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:minus/>
								<ml:apply>
									<ml:neg/>
									<ml:apply>
										<ml:sqrt/>
										<ml:apply>
											<ml:minus/>
											<ml:id xml:space="preserve">z</ml:id>
											<ml:real>3</ml:real>
										</ml:apply>
									</ml:apply>
								</ml:apply>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:matrix>
					</ml:symResult>
				</ml:symEval>
			</math>
			<rendering item-idref="29"/>
		</region>
		<region region-id="2290" left="6" top="3870.75" width="312" height="28.5" align-x="6" align-y="3882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can assign the result of this symbolic evaluation to a function for further numerical use:</p>
			</text>
		</region>
		<region region-id="2628" left="18" top="3912" width="258.75" height="41.25" align-x="53.25" align-y="3936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:function>
						<ml:id xml:space="preserve">sol</ml:id>
						<ml:boundVars>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:boundVars>
					</ml:function>
					<ml:symEval style="default" hide-keywords="false" hide-lhs="false">
						<ml:apply>
							<ml:equal/>
							<ml:apply>
								<ml:minus/>
								<ml:id xml:space="preserve">z</ml:id>
								<ml:real>3</ml:real>
							</ml:apply>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
										<ml:plus/>
										<ml:id xml:space="preserve">x</ml:id>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:command>
							<ml:sequence>
								<ml:id xml:space="preserve">solve</ml:id>
								<ml:id xml:space="preserve">x</ml:id>
							</ml:sequence>
						</ml:command>
						<ml:symResult>
							<ml:matrix rows="2" cols="1">
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:sqrt/>
										<ml:apply>
											<ml:minus/>
											<ml:id xml:space="preserve">z</ml:id>
											<ml:real>3</ml:real>
										</ml:apply>
									</ml:apply>
									<ml:real>2</ml:real>
								</ml:apply>
								<ml:apply>
									<ml:minus/>
									<ml:apply>
										<ml:neg/>
										<ml:apply>
											<ml:sqrt/>
											<ml:apply>
												<ml:minus/>
												<ml:id xml:space="preserve">z</ml:id>
												<ml:real>3</ml:real>
											</ml:apply>
										</ml:apply>
									</ml:apply>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:matrix>
						</ml:symResult>
					</ml:symEval>
				</ml:define>
			</math>
			<rendering item-idref="30"/>
		</region>
		<region region-id="2638" left="18" top="4004.25" width="102" height="36.75" align-x="54" align-y="4026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">sol</ml:id>
						<ml:real>8</ml:real>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:matrix rows="2" cols="1">
							<ml:real>0.23606797749978981</ml:real>
							<ml:real>-4.23606797749979</ml:real>
						</ml:matrix>
					</result>
				</ml:eval>
			</math>
			<rendering item-idref="31"/>
		</region>
		<region region-id="2635" left="150" top="4004.25" width="102" height="36.75" align-x="186" align-y="4026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:id xml:space="preserve">sol</ml:id>
						<ml:real>5</ml:real>
					</ml:apply>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<ml:matrix rows="2" cols="1">
							<ml:real>-0.58578643762690485</ml:real>
							<ml:real>-3.4142135623730949</ml:real>
						</ml:matrix>
					</result>
				</ml:eval>
			</math>
			<rendering item-idref="32"/>
		</region>
		<region region-id="2288" left="6" top="4076.25" width="264" height="48" align-x="17.25" align-y="4086" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click after the last <b>z</b> in the expression below, click <b>solve</b> on the <b>Symbolic</b> toolbar, type a comma, and enter z in the placeholder to the right of the solve keyword. Then press<b> [Enter]</b>
					<inlineAttr font-weight="normal">.</inlineAttr>
				</p>
			</text>
		</region>
		<region region-id="2624" left="18" top="4134.75" width="30" height="15.75" align-x="31.5" align-y="4146" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">a</ml:id>
					<ml:real>1</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="33"/>
		</region>
		<region region-id="2645" left="90" top="4134.75" width="30.75" height="15.75" align-x="104.25" align-y="4146" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">b</ml:id>
					<ml:real>4</ml:real>
				</ml:define>
			</math>
			<rendering item-idref="34"/>
		</region>
		<region region-id="2649" left="162" top="4134.75" width="30" height="15.75" align-x="175.5" align-y="4146" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30">
					<originRef doc-id="EC277587-6E18-4C63-A5C9-882CAF242C09" version-id="C752B91B-5943-4124-A72D-5CD94488A625" branch-id="00000000-0000-0000-0000-000000000000" revision-num="634860124" is-modified="false" region-id="2201" href="C:\Mathcad 13\Documentation\ResourceCenter\EN\qsheet\tutorial\Primers\tu_07b.xmcd" xmlns="http://schemas.mathsoft.com/provenance10">
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					</originRef>
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						<hash/>
					</parentRef>
					<comment xmlns="http://schemas.mathsoft.com/provenance10"/>
					<originComment xmlns="http://schemas.mathsoft.com/provenance10"/>
					<contentHash xmlns="http://schemas.mathsoft.com/provenance10">9d32b6c84f94c672baf9448808df03c6</contentHash>
					<ml:define warning="WarnRedefinedBIUnit">
						<ml:id xml:space="preserve">c</ml:id>
						<ml:real>3</ml:real>
					</ml:define>
				</ml:provenance>
			</math>
			<rendering item-idref="35">
				<element-image-map>
					<box left="1.5" top="0.75" width="28.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/>
				</element-image-map>
			</rendering>
		</region>
		<region region-id="2648" left="18" top="4168.5" width="42.75" height="33.75" align-x="39.75" align-y="4188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:apply xmlns:ml="http://schemas.mathsoft.com/math30">
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							<ml:id xml:space="preserve">a</ml:id>
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						</ml:apply>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:id xml:space="preserve">b</ml:id>
						<ml:id xml:space="preserve">z</ml:id>
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		<region region-id="2686" left="6" top="4226.25" width="20.25" height="12" align-x="13.5" align-y="4236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2682" left="24" top="4242" width="144" height="77.25" align-x="116.25" align-y="4284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">c</ml:id>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">a</ml:id>
								<ml:id xml:space="preserve">z</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">b</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
						</ml:apply>
					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
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					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:div/>
									<ml:apply>
										<ml:sqrt/>
										<ml:real>3</ml:real>
									</ml:apply>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:div/>
								<ml:apply>
									<ml:sqrt/>
									<ml:real>3</ml:real>
								</ml:apply>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:matrix>
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		<region region-id="2386" left="6" top="4334.25" width="259.5" height="24" align-x="16.5" align-y="4344" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Change the value of b, above, by backspacing over the 4 and typing 1/2.</p>
			</text>
		</region>
		<region region-id="2387" left="6" top="4370.25" width="20.25" height="12" align-x="13.5" align-y="4380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
		</region>
		<region region-id="2687" left="30" top="4396.5" width="33.75" height="33.75" align-x="44.25" align-y="4416" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">b</ml:id>
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						<ml:div/>
						<ml:real>1</ml:real>
						<ml:real>2</ml:real>
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		<region region-id="2690" left="30" top="4446" width="141" height="41.25" align-x="122.25" align-y="4470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">c</ml:id>
							<ml:apply>
								<ml:mult/>
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								<ml:id xml:space="preserve">z</ml:id>
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						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">b</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
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					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
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					</ml:command>
					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:sqrt/>
									<ml:real>6</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:sqrt/>
								<ml:real>6</ml:real>
							</ml:apply>
						</ml:matrix>
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			<rendering item-idref="39"/>
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		<region region-id="2389" left="6" top="4508.25" width="259.5" height="24" align-x="16.5" align-y="4518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Change the value of a, above, by backspacing over the 1 and typing -3.</p>
			</text>
		</region>
		<region region-id="2390" left="6" top="4550.25" width="20.25" height="12" align-x="13.5" align-y="4560" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
			</text>
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		<region region-id="2692" left="30" top="4584.75" width="36.75" height="15.75" align-x="43.5" align-y="4596" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve">a</ml:id>
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		<region region-id="2693" left="198" top="4626.75" width="122.25" height="28.5" align-x="204.75" align-y="4638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The solutions are complex numbers. </p>
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			<math optimize="false" disable-calc="false">
				<ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:equal/>
						<ml:apply>
							<ml:div/>
							<ml:id xml:space="preserve">c</ml:id>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve">a</ml:id>
								<ml:id xml:space="preserve">z</ml:id>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve">b</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
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					</ml:apply>
					<ml:command>
						<ml:sequence>
							<ml:id xml:space="preserve">solve</ml:id>
							<ml:id xml:space="preserve">z</ml:id>
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					<ml:symResult>
						<ml:matrix rows="2" cols="1">
							<ml:apply>
								<ml:neg/>
								<ml:apply>
									<ml:mult/>
									<ml:apply>
										<ml:sqrt/>
										<ml:real>2</ml:real>
									</ml:apply>
									<ml:imag symbol="i">1</ml:imag>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:apply>
									<ml:sqrt/>
									<ml:real>2</ml:real>
								</ml:apply>
								<ml:imag symbol="i">1</ml:imag>
							</ml:apply>
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		<region region-id="2309" left="6" top="4686.75" width="312" height="28.5" align-x="6" align-y="4698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">For further information and examples, see the QuickSheets on <b>
						<u>
							<link href=".\..\..\Symbolics\symbolictoc.xmcd" title="Links to section" popup="false">Symbolic Math</link>
						</u>
					</b>.</p>
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