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 is taken as = x

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

As per the question,

x – 8x = 180

⇒ x – 8x – 180 = 0

⇒ x – 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 a smaller number = 144

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

The numbers are 12 and 18

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