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.

Let the larger number = x
Let the smaller number = y
Therefore, y² = 8x
According to the question, x² – y² = 180
⇒ x² – 8x = 180 [ As, y² = 8x ]
⇒ 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
Either x = 18 or x = – 10
But, x ≠ – 10, as x larger of the two numbers So, x = 18
Therefore, the larger number = 18
Hence the smaller number = y = √144 = 12