Question

Find two consecutive positive even integers whose product is 224.

Answer 1

Let the two consecutive positive even integers be $$x$$ and $$(x + 2)$$.

According to the given condition: $$x(x + 2) = 224$$

$$x^2 + 2x - 224 = 0$$

Factoring the quadratic equation: $$x^2 + 16x - 14x - 224 = 0$$

$$x(x + 16) - 14(x + 16) = 0$$

$$(x + 16)(x - 14) = 0$$

$$x = -16 \quad \text{or} \quad x = 14$$ 

Since the integers are positive, reject $$x = -16$$.

Therefore, $$x = 14$$ and the next consecutive even integer is $$14 + 2 = 16$$.

Hence, the required integers are 14 and 16.