Question

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

Answer 1

SOLUTION :

Let x be the larger number & square of a smaller number be 8x .

A.T.Q

Square of a larger number - Square of a smaller number = 180

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

Case 1 :  

If x = 18 , then  square of a smaller number = 8x = 8 × 18 = 144  

Smaller number = √144 = ± 12

Case 2 :

If x = - 10 , then  square of a smaller number = 8x = 8 × -10 = - 80

This is not possible as square of a number cannot be negative.  

Hence, the required number are (18 , 12) & (18 , - 12) .

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Answer 2

Let the larger number = x
Then the square of the smaller number = 8 times the larger number = 8x
and the square of the larger numbe r = x
According to 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 smaller number = 144
Hence, the smaller number = sqrt(144) = 12
The numbers are 12 and 18
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