# 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.

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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.

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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