Question

Find two consecutive integers such that the sum of their squares is 61

Answer 1

Let first integer = x
Then second integer = x + 1
According to the condition,
(x)² + (x + 1)² = 61
⇒ x² + x² + 2x + 1 = 61
⇒ 2x² + 2x + 1 - 61 = 0
⇒ 2x² + 2x - 60 = 0
⇒ x² + x - 30 = 0   ...(Dividing by 2)
⇒ x² + 6x - 5x - 30 = 0
⇒ x(x + 6) -5(x + 6) = 0
⇒ (x + 6)(x - 5) = 0
Either x + 6 = 0,
then x = -6
or
x - 5 = 0,
then x = 5
(i) If x = -6, then
First integer = -6
and second = -6 + 1 = -5
(ii) If x = 5, then
First integer = 5
and second = 5 + 1 = 6
∴ Required integers are (-6, -5),(5, 6).