If the product of two positive consecutive even integers is 288, find the integers.

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

Given, product of two consecutive even integers = 288

$\Rightarrow x(x + 2) = 288 \\[1em] \Rightarrow x^2 + 2x = 288 \\[1em] \Rightarrow x^2 + 2x - 288 = 0 \\[1em] \Rightarrow x^2 + 18x - 16x - 288 = 0 \\[1em] \Rightarrow x(x + 18) - 16(x + 18) = 0 \\[1em] \Rightarrow (x - 16)(x + 18) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow x - 16 = 0 \text{ or } x + 18 = 0 \text{ (Zero-product rule) } \\[1em] \Rightarrow x = 16 \text{ or } x = -18$

Since the numbers are natural number so x ≠ -18.

∴ x = 16 , x + 2 = 18.

Hence, required integers are 16, 18.