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This question already has an answer here:

$$0 = a(x^2 + \frac ba x) + c = a(x^2 + \frac ba x + \frac{b^2}{4a^2}) -\frac{b^2}{4a} + c$$ $$= a(x + \frac b{2a})^2 + c - \frac{b^2}{4a}$$
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 $a(x^2 + \frac bax) + c$?

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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

There's no perfect way of doing something in Mathematics, but we're just interested in interesting ways to do things and I guess that's all Mathematics is all about.

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