The difference between squares of two numbers is 120. The square of sm

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

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

Answer»

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.