Question

Two integers differ by 2 and sum of their squares is 52. The integers are ______.

Options

• 4 and 6

• 4 or 6

• – 4 or 6

• – 4 and – 6 or 6 and 4

Answer 1

Two integers differ by 2 and sum of their squares is 52. The integers are – 4 and – 6 or 6 and 4.

Explanation:

Let the required integers are x and x + 2 and sum of their squares is 52.

∴ x² + (x + 2)² = 52

`\implies` x² + x² + 4x + 4 – 52 = 0

`\implies` 2x² + 4x – 48 = 0

`\implies` 2(x² + 2x – 24) = 0

`\implies` x² + 2x – 24 = 0

`\implies` x² + 6x – 4x – 24 = 0

`\implies` x(x + 6) – 4(x + 6) = 0

`\implies` (x – 4)(x + 6) = 0

Either x – 4 = 0 or x + 6 = 0

`\implies` x = 4 or x = – 6

`\implies` x = 4 or – 6

When x = 4; required integers are 4 and 4 + 2 i.e. 6

When x = – 6; required integers are – 6 and – 6 + 2

i.e. – 6 and – 4