Find two consecutive even natural numbers such that the sum of their squares is 340.

Let the required two consecutive even integers be x , x + 2

Given, sum of squares of two consecutive even natural numbers = 340

$\Rightarrow x^2 + (x + 2)^2 = 340 \\[1em] \Rightarrow x^2 + x^2 + 4 + 4x = 340 \\[1em] \Rightarrow 2x^2 + 4x + 4 -340 = 0 \\[1em] \Rightarrow 2x^2 + 4x - 336 = 0 \\[1em] \Rightarrow 2(x^2 + 2x - 168) = 0 \\[1em] \Rightarrow x^2 + 2x - 168 = 0 \\[1em] \Rightarrow x^2 + 14x - 12x - 168 = 0 \\[1em] \Rightarrow x(x + 14) - 12(x + 14) = 0 \\[1em] \Rightarrow (x - 12)(x + 14) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow x - 12 = 0 \text{ or } x + 14 = 0 \text{ (Zero-product rule) } \\[1em] \Rightarrow x = 12 \text{ or } x = -14$

Since the numbers are natural number so x ≠ -14

∴ x = 12 , x + 2 = 14

Hence, required natural numbers are 12 , 14 .