the difference of squares of two numbers is 180. the square of smaller no. is 8 times the larger number. find the two numbers .

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^{2}

According to the question,

x^{2} - 8x = 180

=> x^{2} - 8x - 180 = 0

=> x^{2} - 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**