Quadratic Roots

If $\alpha$ & $\beta$ are roots of a quadratic then,

$x=\alpha$ & $x=\beta$

so,

$x-\alpha =0$ & $x-\beta =0$

Multiplying these two to get a quadratic,

$$(x-\alpha )(x-\beta )=0$$

Multiplying out the brackets and simplifying,

$$x^2-(\alpha +\beta )x+\alpha \beta =0$$

Scaling by $\gamma$,

$$\gamma x^2-\gamma (\alpha +\beta )x+\alpha \beta \gamma =0$$

Then comparing coefficients with,

$$a{x}^2+bx+c=0$$

We find,

$a=\gamma$,   $b=-\gamma (\alpha +\beta )$,  &  $c=\alpha \beta \gamma$.