Question

Two integers differ by 15 and the sum of their squares is 1225. Determine the integers.

Answer 1

Let's call the two integers x and y. Since they differ by 15, we can write an equation as y = x - 15.

The sum of their squares is 1225, so we can write another equation as x^2 + y^2 = 1225.

Substituting y with x - 15 in the second equation, we have:
x^2 + (x - 15)^2 = 1225.

Expanding the equation:
x^2 + (x - 15)(x - 15) = 1225,
x^2 + (x^2 - 30x + 225) = 1225,
2x^2 - 30x - 1000 = 0.

Dividing the equation by 2:
x^2 - 15x - 500 = 0.

This equation can be factored as (x - 25)(x + 20) = 0.

So, x - 25 = 0 or x + 20 = 0.
If x - 25 = 0, then x = 25.
If x + 20 = 0, then x = -20.

If x = 25, then y = x - 15 = 25 - 15 = 10.
If x = -20, then y = x - 15 = -20 - 15 = -35.

Therefore, the two integers are 25 and 10 or -20 and -35.