#### Question: The difference between squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

#### Video Explanation

#### Explanatory Answer

Step 1: Frame equation from information given in the statements

Let the larger number be L.

Let the smaller number be S.

Statement 1: Difference between squares of the two numbers is 180.

i.e., L^{2} -S^{2} = 180 ------------(1)

Statement 2: Square of the smaller number is 8 times the larger number.

i.e., S^{2} = 8L ----------(2)

Substitute S^{2} = 8L in (1)

L^{2} - 8L = 180

L^{2} - 8L - 180 = 0

##### Step 2: Factorize the quadratic equation to find L

L^{2} - 18 L + 10 L - 180 = 0

L(L - 18) + 10(L - 18) = 0

(L - 18)(L + 10) = 0

L = 18 and L = -10

If L = 18, S^{2} = 8 * 18 = 144

S = 12 or - 12

If L = -10, S^{2} = 8 * (-10) = -80

Square of real number cannot be negative

∴ L = 18 and S = 12; L = 18 and S = -12