SOLUTION: Seperate 30 into parts so that the sum of their squares is a minimum

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Question 893165: Seperate 30 into parts so that the sum of their squares is a minimum
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x%2By=30 and interest in the expression x%5E2%2By%5E2.

y=30-x.
y%5E2=%2830-x%29%5E2
900-30x-30x%2Bx%5E2
900-60x%2Bx%5E2
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The sum of squares expression is
x%5E2%2B%28x%5E2-60x%2B900%29
2x%5E2-60x%2B900-------This does have a minimum.

2%28x%5E2-15x%2B450%29
highlight%282%28x%2B15%29%28x-30%29%29
The roots are -15 and 30.
The minimum value of 2%28x%2B15%29%28x-30%29 will occur in the exact middle of these roots.
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%28-15%2B30%29%2F2=15%2F2=7%261%2F2, for x.

The minimum value for the sum of those squares will be for highlight%28x=15%2F2%29 and highlight%28y=30-7%261%2F2=22%261%2F2%29.