Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Answer 1

Let the larger number = x

Then the square of the smaller number = 8 times the larger number = 8x

and the square of the larger number = x²

According to the question,

  x² - 8x = 180

  =>  x² - 8x - 180 = 0

  =>  x² - 18x + 10x - 180 = 0

  =>  x(x - 18) + 10(x - 18) = 0

  =>  (x - 18) (x + 10)  = 0

  =>  x - 18 = 0  or x + 10 = 0

  =>  x = 18  or x = -10

Thus, the larger number = 18 or -10

Then, the square of the smaller number  = 8(18)  or     8(-10)

  = 144       or     -80

The square of a number can't be negative, so, the square of smaller number = 144

Hence, the smaller number = sqrt(144) = 12

The numbers are 12 and 18