The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger number, find the numbers.
Let the positive integers be ‘a’ and ‘b’.
Given, difference of the squares of two positive integers is 180.
⇒ a2 – b2 = 180
Also, square of the smaller number is 8 times the larger.
⇒ b2 = 8a
Thus, a2 – 8a – 180 = 0
⇒ a2 – 18a + 10a – 180 = 0
⇒ a(a – 18) + 10(a – 18) = 0
⇒ (a + 10)(a – 18) = 0
⇒ a = -10, 18
Thus, the other number is
324 – 180 = b2
⇒ b = 12
Numbers are 12, 18