Question

The sum of the squares of two consecutive integers is 41. The integers are ______.

Options

• 4 and – 5 or – 4 and 5

• 4 and 5 or – 4 and – 5

• 3 and 4 or – 4 and – 3

• 6 and 3 or – 6 and – 3

Answer 1

The sum of the squares of two consecutive integers is 41. The integers are 4 and 5 or – 4 and – 5.

Explanation:

Let the consecutive integers are x and x + 1.

Then, according to given condition, we have

x² + (x + 1)² = 41

`\implies` x² + x² + 2x + 1 = 41

`\implies` 2x² + 2x + 1 – 41 = 0

`\implies` 2x² + 2x – 40 = 0

`\implies` 2(x² + x – 20) = 0

`\implies` x² + x – 20 = 0

`\implies` x² + 5x – 4x – 20 = 0

`\implies` x(x + 5) – 4(x + 5) = 0

`\implies` (x + 5)(x – 4) = 0

Either x + 5 = 0 or x – 4 = 0

`\implies` x = – 5 or 4

When x = – 5, thus consecutive integers are – 5 and – 5 + 1

i.e. – 5 and – 4

When x = 4

∴ Consecutive integers are 4 and 4 + 1

i.e. 4 and 5