Solution 2.3:6a

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{{NAVCONTENT_START}}
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Using the squaring rule, we recognize the polynomial as the expansion of
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<center> [[Image:2_3_6a.gif]] </center>
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<math>\left( x-1 \right)^{2}</math>,
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{{NAVCONTENT_STOP}}
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<math>x^{2}-2x+1=\left( x-1 \right)^{2}</math>
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This quadratic expression has its smallest value, zero, when
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<math>x-\text{1}=0</math>, i.e.
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<math>x=\text{1}</math>. All non-zero values of
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<math>x-\text{1}</math>
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give a positive value for
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<math>\left( x-1 \right)^{2}</math>.
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NOTE: If we draw the curve
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<math>y=\left( x-1 \right)^{2}</math>, we see that it has a minimum value of zero at
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<math>x=\text{1}</math>.
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[[Image:2_3_6_a.gif|center]]
[[Image:2_3_6_a.gif|center]]

Revision as of 10:51, 21 September 2008

Using the squaring rule, we recognize the polynomial as the expansion of \displaystyle \left( x-1 \right)^{2},


\displaystyle x^{2}-2x+1=\left( x-1 \right)^{2}


This quadratic expression has its smallest value, zero, when \displaystyle x-\text{1}=0, i.e. \displaystyle x=\text{1}. All non-zero values of \displaystyle x-\text{1} give a positive value for \displaystyle \left( x-1 \right)^{2}.

NOTE: If we draw the curve \displaystyle y=\left( x-1 \right)^{2}, we see that it has a minimum value of zero at \displaystyle x=\text{1}.


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