Question

Find two consecutive natural numbers such that the sum of their squares is 61.

Answer 1

Let the first natural number = x
then second natural number = x + 1
According to the condition, (x)² + (x + 1)² = 61
⇒ x² + x² + 2x + 1 - 61 = 0
⇒ 2x² + 2x - 60 = 0
⇒ x² + x - 30 = 0
⇒ 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
∴ The number are positive.
∴ x = -6 is not possible
Hence the first natural number = 5
and secod natural number = 5 + 1 = 6.