# Thread: Perfect Square

1. ## Perfect Square

Determine all integers such that $x^2 + 19x + 92$

is a square.

2. Let $=k^2$

$x^2+19x+92-k^2=0$

Apply quadratic formula: $x=\frac{-19\pm{\sqrt{19^2-4(92-k^2)}}}{2}$
$x=\frac{-19\pm{\sqrt{4k^2-7}}}{2}$

Determine integers such that $4k^2-7=m^2$
$4k^2-m^2=7$
$(2k+m)(2k-m)=7$
As k and m are integers, these 2 factors must be factors of 7

$2k+m=7$
$2k-m=1$
$4k=8\rightarrow{k}=2$

Insert k into the equation to receive your answer

$x=\frac{-19\pm3}{2}$