7. 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 us say, the larger and smaller number be x and y respectively. As per the question given, x2 – y2 = 180 and y2 = 8x ⇒ 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, -10 However, the larger number cannot considered as negative number, as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible. Therefore, the larger number will be 18 only. x = 18 ∴ y2 = 8x = 8 × 18 = 144 ⇒ y = ±√144 = ±12 ∴ Smaller number = ±12 Therefore, the numbers are 18 and 12 or 18 and -12.