Question

The difference between squares of two numbers is 120. The square of smaller number is twice the greater number. Find the numbers.

Answer 1

Let the numbers be x and y (x > y).

According to the given condition,

x² – y² = 120 …(i) 

y² = 2x …(ii)

Substituting y² = 2x in equation (i), we get

x² – 2x = 120 

∴ x² – 2x – 120 = 0 

∴ x² – 12x + 10x – 120 = 0 

∴ x(x – 12) + 10(x – 12) = 0 

∴ (x – 12)(x + 10) = 0

By using the property, if the product of two numbers is zero, then at least one of them is zero, we get

∴ x – 12 = 0 or x + 10 = 0 

∴ x = 12 or x = -10

But x ≠ -10

as, y² = 2x = 2(-10) = -20 …[Since, the square of number cannot be negative]

∴ x = 12

Smaller number = y² =2x

∴ y² = 2 × 12

∴ y² = 24

∴ y = ± √24 …[Taking square root of both sides]

∴ The smaller number is √24 and greater number is 12 or the smaller number is – √24 and greater number is 12.