The difference of two positive integers is 3 and the sum of their squares is 89. The integers are _____?
Answer: {5, 8}
Explanation
Let's denote the two integers as x and y, where x > y:
Given x - y = 3 and x² + y² = 89.
From x - y = 3, we get x = y + 3.
Substitute x in the second equation:
(y + 3)² + y² = 89
y² + 6y + 9 + y² = 89
2y² + 6y - 80 = 0
y² + 3y - 40 = 0
(y + 8)(y - 5) = 0
y = -8 (not possible since y is positive) or y = 5.
If y = 5, then x = y + 3 = 8.
Check: 8² + 5² = 64 + 25 = 89.
The integers are 8 and 5.
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