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,
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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{{Displayed math||<math>x^{2}-4x+2 = (x-2)^{2}-2^{2}+2 = (x-2)^{2}-2\,\textrm{.}</math>}}
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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 <math>(x-2)^{2}</math> is a quadratic, this term is always larger than or equal to 0 and the whole expression is therefore at least equal to -2, which occurs when <math>x-2=0</math> and the quadratic is zero, i.e. <math>x=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>.
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Current revision

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 = (x-2)^{2}-2^{2}+2 = (x-2)^{2}-2\,\textrm{.}

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

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