Po-Shen Loh in his famous video shows how to systematically find quadratic equation's roots.
He find the roots for following quadratic equation.
$x^2-8x+12 = 0$
Product: 12, Sum: 8
He divides sum by half i.e. 8/2 = 4
Proceeds to find roots: $4 - u$ & $4 + u.$
$ (4 - u) * (4 + u) = 12$
$16 - u^2 = 12 $
$16 - 12 = u^2$
$ 4 = u^2$
$u = \pm2$
Now, if we substitute value of $u$ in $4 - u$ & $4 + u$, we get, $2$ and $6$
But, this methods doesn't yield result for $x^2-x-132 = 0$
Let's divide sum by half i.e. $(1) * 1/2 = 1/2$
Proceeds to find roots: $1/2-u$ & $1/2+ u$
$ (1/2 - u) * (1/2 + u) = - 132$
$1/4 - u^2 = - 132 $
$1/4 + 132 = u^2$
$ 529/4 = u^2$
$ 132.25 = u^2$
$u = ?$
How to systematically find roots of $x^2-x-132 = 0$ with above method?
$$u^2=132.25$$
$$u=\pm11.5$$
$$\frac{1}{2}+11.5=12$$
$$\frac{1}{2}-11.5=-11$$
$$(x-12)(x+11)=0$$
The $-132$ does change things, by the way.