The square of the sum of two given natural numbers is 784, while the product of the two given numbers is 192. Find the positive difference between the squares of these two given numbers.

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 20 Jul 2023 Shift 1)

Answer 1

Let the numbers are X and Y

Given:

(X + Y)2 = 784 and XY = 192

Calculation:

(X + Y)2 = 784 ⇒ (X + Y) = 28

⇒ X² + Y² + 2XY = 784

⇒ X2 + Y2 + 2 × 192 = 784

⇒ X2 + Y2 = 400

So,

⇒ X2 + Y2 - 2XY = 400 - 2 × 192

⇒ X2 + Y2 - 2XY = 16

⇒ (X - Y)² = 16

⇒ X - Y = 4

Now,

X2 - Y2 = (X + Y)(X - Y)

⇒ 28 × 4 = 112

∴ The correct option is 4

Alternate MethodAccording to the question,

(x + y)2 = 784

⇒ x + y = √784 = 28

xy = 192

x - y = √{(x + y)2 - 4xy}

⇒ √{282 - 4 × 192} = √16 = 4

Now, x2 - y2 = (x + y)(x - y) = 28 × 4 = 112