Given
Two positive integers x and y
x - y = 4
x² + y = 68
Find
x and y
Solution
Add the two equations together.
... (x - y) + (x² + y) = (4) + (68)
... x² + x = 72
Rearrange to standard form and factor.
... x² + x - 72 = 0
... (x + 9)(x - 8) = 0
Use the zero product rule to find the solutions. That rule says the product is zero when one or more factors is zero.
... x + 9 = 0 ⇒ x = -9
... x - 8 = 0 ⇒ x = 8 . . . . . . the positive solution
Then we can find y from
... 8 - y = 4
... y = 4 . . . . . . . add y-4 to the equation
The two positive integers are 8 and 4.