Solution 2.3:6b

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By completing the square, the second degree polynomial can be rewritten as a quadratic plus a constant, and then it is relatively straightforward to read off the expression's minimum value,
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<center> [[Image:2_3_6b.gif]] </center>
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<math>x^{2}-4x+2=\left( x-2 \right)^{2}-2^{2}+2=\left( x-2 \right)^{2}-2</math>
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Because
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<math>\left( x-2 \right)^{2}</math>
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is a quadratic, this term is always larger than or equal to
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<math>0</math>
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and the whole expression is therefore at least equal to
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<math>-\text{2}</math>, which occurs when
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<math>x-\text{2}=0\text{ }</math>
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and the quadratic is zero, i.e.
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<math>x=\text{2}</math>.

Revision as of 10:56, 21 September 2008

By completing the square, the second degree polynomial can be rewritten as a quadratic plus a constant, and then it is relatively straightforward to read off the expression's minimum value,


\displaystyle x^{2}-4x+2=\left( x-2 \right)^{2}-2^{2}+2=\left( x-2 \right)^{2}-2


Because \displaystyle \left( x-2 \right)^{2} is a quadratic, this term is always larger than or equal to \displaystyle 0 and the whole expression is therefore at least equal to \displaystyle -\text{2}, which occurs when \displaystyle x-\text{2}=0\text{ } and the quadratic is zero, i.e. \displaystyle x=\text{2}.

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