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 smaller number be x

According to the given condition,

(smaller no.)² = 2(greater no.)

∴ `x^2/2` = Greater no.

According to the given condition,

`(x^2/2)^2 - x^2 = 120`

∴ `x^4/4 - x^2 = 120`

∴ `(x^4 - 4x^2)/4 = 120`

∴ x⁴ - 4x² = 480

∴ x⁴ − 4x² − 480 = 0

∴ `(x^2)^2 - 4x^2 - 480 = 0`

let, x² = a

∴ a² − 4a − 480 = 0

∴ a² − 24a + 20a - 480 = 0

∴ a (a − 24) + 20 (a − 24) = 0

∴ (a − 24) (a + 20) = 0

∴ a = 24 or a = −20

Resubstituting a = x²

∴ x² = 24 or x² = −20

Here,

x² = −20 is rejected because the square of a no. cannot be negative.

∴ x² = 24

∴ Taking square root on both sides

∴ x = ± `sqrt24`

∴ Greater no. = `x^2/2`

= `(± sqrt 24)^2/2`

= `24/2`

= 12

∴ The smaller number is `-sqrt 24` & greater number is 12.

or The smaller number is `sqrt 24` & greater number is −12.