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Solution 2.3:6a

From Förberedande kurs i matematik 1

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

Current revision

Using the rule (a+b)2=a2+2ab+b2, we recognize the polynomial as the expansion of (x1)2,

x22x+1=(x1)2.

This quadratic expression has its smallest value, zero, when x1=0, i.e. x=1. All non-zero values of x1 give a positive value for (x1)2.


Note: If we draw the curve y=(x1)2, we see that it has a minimum value of zero at x=1.