Quadratic Equation to its binomial form

Question

How do I convert a quadratic equation to its binomial form

For example how does $x^2 - 12x - 13$ become $(x-13)(x+1)$ ?

Answer 1

if your equation is always of the form $x^2 + bx + c$ then there are unique (possibly complex) numbers $\alpha,\beta$ such that $x^2 + bx + c = (x-\alpha)(x-\beta)$. In particular, $\alpha$ and $\beta$ can be found using the quadratic formula. $$ \alpha = \frac{-b + \sqrt{b^2 - 4c}}{2} ~~~~\text{and}~~~~ \beta = \frac{- b - \sqrt{b^2 - 4c}}{2} $$

Answer 2

You can apply the quadratic formula. $$x = \frac{-b \pm \sqrt{b^2 - 4 ac}}{2a}$$ We name the two answers we get (from the plus/minus), $$p = \frac{-b + \sqrt{b^2 - 4 ac}}{2a} \quad \quad q = \frac{-b - \sqrt{b^2 - 4 ac}}{2a}$$ Then we can prove with some messy algebra, $$a(x-p)(x-q) = ax^2 + bx + c$$