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

**Solution:**

Let the larger number be x.

The square of the smaller number = 8 times the larger number = 8x.

Given that the difference of squares of the two numbers is 180.

Square of larger number - Square of smaller number = 180

**⇒** x^{2} - 8x = 180

**⇒** x^{2} - 8x - 180 = 0

Now, we will solve this quadratic equation by factoring.

**⇒** x^{2} - 18x + 10x - 180 = 0

**⇒** x(x -18) + 10(x - 18) = 0

**⇒** (x -18)(x + 10) = 0

**⇒** x - 18 = 0 and x + 10 = 0

**⇒** x = 18 and x = - 10

**Case 1:** If the larger number is 18, then square of smaller number = 8 × 18

Therefore, smaller number = ± √8 × 18

= ± √ (__2 × 2 × 2__ × __3 × 3 × 2__)

= ± (2 × 2 × 3)

= ± 12

**Case 2:** If larger number is - 10, then square of smaller number = 8 × (- 10) = - 80

The square of any number cannot be negative.

∴ x = -10 is not applicable.

Thus, the numbers are 18, 12 (or) 18, - 12.

**☛ Check:** NCERT Solutions for Class 10 Maths Chapter 4

**Video Solution:**

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

Class 10 Maths NCERT Solutions Chapter 4 Exercise 4.3 Question 7

**Summary:**

The difference of squares of the two numbers is 180. If the square of a smaller number is 8 times the larger number then the two numbers are 12, 18 or -12, 18.

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