The problem is involves quadratic equation.
x = larger positive integer
y = smaller positive integer
x - y = 7 (the difference between two positive numbers)
- y = 7 - x
y = x - 7
y = x - 7 (smaller positive number in terms of x)
[x + (x-7)]² = 289 (the square of the sum of the two numbers)
(2x - 7)² = 289
4x² - 14x - 14x + 49 = 289
4x² -28x + 49 - 289 = 0
4x² - 28x - 240 = 0
4 (x² - 7x - 60) = 0
Factor x² -7x - 60:
(x - 12) (x + 5) = 0
x - 12 = 0
x = 12
x + 5 = 0
x = -5
Choose the positive value, x = 12.
Substitute 12 to x:
Larger number: x = 12
Smaller number : x-7 = 12 - 7 = 5
ANSWER: The numbers are 12 and 5.
Check:
The difference of two positive numbers is 7.
12 - 5 = 7
7 = 7 (true)
The square of their sum is 289.
(12 + 5)² = 289
(17)² = 289
289 = 289 (true)
