Question

Find three successive even natural numbers, the sum of whose squares is 308.

Answer 1

 Let first even number = 2x
second even number = 2x + 2
third even number = 2x + 4
Now according to the condition,
(2x)² + (2x + 2)² + (2x + 4)² = 308
⇒ 4x² + 4x² + 8x + 4 + 4x² + 16x + 16 = 308
⇒ 12x² + 24x + 20 - 308 = 0
⇒ 12x² + 24x - 288 = 0
⇒ x² + 2x - 24 = 0    ...(Dividing y 12)
⇒ x² + 6x - 4x - 24 = 0
⇒ x(x + 6) -4(x + 6) = 0
⇒ (x + 6)(x - 4) = 0
Either x + 6 = 0,
then x = -6
But i is not a natural number, hence not possible.
or
x - 4 = 0,
then x = 4
∴ First even natural number = 2x = 2 x 4 = 8
second number = 8 + 2 = 10
and the third number = 10 + 2 = 12.