SOLUTION: Find two numbers whose sum is 17 and whose squares total 145

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Question 389639: Find two numbers whose sum is 17 and whose squares total 145
Answer by richard1234(7193) About Me  (Show Source):
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The easiest way is probably to guess and check, since it is likely that both numbers will be integers. Guessing and checking, the numbers are 9 and 8.

Or, we can use p+%2B+q+=+17 and p%5E2+%2B+q%5E2 = 145. Squaring the first equation,

%28p+%2B+q%29%5E2+=+p%5E2+%2B+2pq+%2B+q%5E2+=+289 --> 2pq+=+144 --> pq+=+72.

If we assume that p and q are roots of a polynomial x%5E2+%2B+bx+%2B+c, use Vieta's formula:

The sum of the roots is -b --> p+%2B+q+=+17+=+-b
The product of the roots is c --> pq+=+72+=+c

This implies b = -17 and c = 72, so p and q are roots of the polynomial x%5E2+-+17x+%2B+72. The polynomial can be factored as %28x+-+8%29%28x+-+9%29 so p,q = (8,9).