Question 893165
{{{x+y=30}}} and interest in the expression {{{x^2+y^2}}}.


{{{y=30-x}}}.
{{{y^2=(30-x)^2}}}
{{{900-30x-30x+x^2}}}
{{{900-60x+x^2}}}
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The sum of squares expression is
{{{x^2+(x^2-60x+900)}}}
{{{2x^2-60x+900}}}-------This does have a minimum.


{{{2(x^2-15x+450)}}}
{{{highlight(2(x+15)(x-30))}}}

The roots are -15 and 30.
The minimum value of {{{2(x+15)(x-30)}}} will occur in the exact middle of these roots.
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{{{(-15+30)/2=15/2=7&1/2}}}, for x.


The minimum value for the sum of those squares will be for {{{highlight(x=15/2)}}} <b>and</b> {{{highlight(y=30-7&1/2=22&1/2)}}}.