SOLUTION: the sum of two integers is 10 and the sum of their squares is 82. find the integers

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Question 539168: the sum of two integers is 10 and the sum of their squares is 82. find the integers
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+a+%2B+b+=+10+
+a%5E2+%2B+b%5E2+=+82+
-----------------
+b+=+10+-+a+
By substitution:
+a%5E2+%2B+%28+10+-+a+%29%5E2+=+82+
+a%5E2+%2B+100+-+20a+%2B+a%5E2+=+82+
+2a%5E2+-+20a+%2B+100+=+82+
+2a%5E2+-+20a+%2B+18+=+0+
+a%5E2+-+10a+%2B+9+=+0+
complete the square
+a%5E2+-+10a+=+-9+
a%5E2+-+10a+%2B+%28-10%2F2%29%5E2+=+-9+%2B+%28-10%2F2%29%5E2+
+a%5E2+-+10a+%2B+25+=+16+
+%28+a+-+5+%29%5E2+=+4%5E2+
Take the square root of both sides
+a+-+5+=+4+
+a+=+9+
It is also true that
+a+-+5+=+-4+
+a+=+1+
If I say that +a+=+9+, then
+a+%2B+b+=+10+
+b+=+1+
The integers are 9 and 1
check:
+9%5E2+%2B+1%5E2+=+82+
+81+%2B+1+=+82+
+82+=+82+
OK