Don't understand the step in Deriving quadratic equation

Question

It's from the book essenstial mathematics for computer graphics fast. I don't understand why after factorize the left side will become step 4. Can anyone explain? Thanks.

Answer 1

I can't comment due to less reputation. Here is a hint $$ (a+b)^2=a^2+2ab+b^2$$ Also here is a proof I like using the relationship between the zeros of a polynomial and its constant coefficients. This relationship is for a quadratic polynomial, $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$, $$ \alpha\beta= \frac {c}{a}$$ $$ \alpha + \beta =\frac {-b}{a}$$ Using this, \begin{align*} (\alpha - \beta)^2 &= (\alpha + \beta)^2 - 4\alpha\beta\\ (\alpha - \beta)^2 &= \left(\frac {-b}{a}\right)^2 -\frac{4c}{a}\\ \alpha - \beta &= \frac {\pm\sqrt {b^2 - 4ac}}{a} \end{align*} Using this and $\alpha + \beta = \frac {-b}{a}$ , one can easily derive the quadratic formula