Question

Find two consecutive even integers such that the square of the second, decreased by twice the first is 52

Answer 1

Let the first integer be $x$. So, the next even number is $x+2$

Let's translate the formula: the square of the second is $(x+2)^2$, twice the first is $2x$. So, the equation is

$(x+2)^2 - 2x = 52$

Expand the square to get

$x^2+4x+4-2x = 52 \iff x^2+2x+4 = 52 \iff x^2+2x - 48 = 0$

The solutions to this equations are

$x = -8,\quad x = 6$

So, the possible consecutive even numbers are

$-8\text{ and } -6\quad\text{or}\quad 6\text{ and } 8$