Let the first integer be [tex] x [/tex]. So, the next even number is [tex] x+2 [/tex]
Let's translate the formula: the square of the second is [tex] (x+2)^2 [/tex], twice the first is [tex] 2x [/tex]. So, the equation is
[tex] (x+2)^2 - 2x = 52 [/tex]
Expand the square to get
[tex] x^2+4x+4-2x = 52 \iff x^2+2x+4 = 52 \iff x^2+2x - 48 = 0 [/tex]
The solutions to this equations are
[tex] x = -8,\quad x = 6 [/tex]
So, the possible consecutive even numbers are
[tex] -8\text{ and } -6\quad\text{or}\quad 6\text{ and } 8 [/tex]
