If the product of two consecutive even integers is 224 , find the integers.

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

Given, product of two consecutive even integers = 224

$\Rightarrow x(x + 2) = 224 \\[1em] \Rightarrow x^2 + 2x = 224 \\[1em] \Rightarrow x^2 + 2x - 224 = 0 \\[1em] \Rightarrow x^2 + 16x - 14x - 224 = 0 \\[1em] \Rightarrow x(x + 16) - 14(x + 16) = 0 \\[1em] \Rightarrow (x + 16)(x - 14) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow x + 16 = 0 \text{ or } x - 14 = 0. \\[1em] \Rightarrow x = -16 \text{ or } x = 14 \\[1em]$

∴ When x = 14 , x + 2 = 16 and when x = -16 , x + 2 = -14.

Hence, required integers are 14 , 16 or -16, -14 .