$$0 = a(x^2 + \frac ba x) + c = a(x^2 + \frac ba x + \frac{b^2}{4a^2}) -\frac{b^2}{4a} + c$$ $$= a(x + \frac b{2a})^2 + c - \frac{b^2}{4a}$$
It is more than obvious that the above equation simplifies to the Quadratic Formula, yet I was curious as to why the method of simplifying is done as seen above from the original
$ax^2 + bx + c = 0$ in that way, for instance how does it become $a(x^2 + \frac bax) + c$?

## marked as duplicate by Jonas Meyer, user147263, apnorton, Daniel Fischer, Adam HughesDec 12 '14 at 22:11

$$ax^2+bx+c=0\tag{1}$$ taking $a$ common from first two terms $$a\left(x^2 + \frac bax\right) + c=0$$