<s>[INST] Proving the quadratic formula (for dummies) I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from $ax^2+bx+c=0$ to $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good at maths, so can someone prove the quadratic formula in a simple way, with no complicated words? All help appreciated.
 [/INST] You should work this way to prove it:
$ax^2+bx+c=0 \quad \iff \quad (\sqrt{a}x + \frac{1}{2}b/\sqrt{a})^2 + (c-\frac{1}{4}b^2/a) = 0$
$\iff \quad (\sqrt{a}x + \frac{1}{2}b/\sqrt{a})= + - \sqrt{\frac{1}{4}b^2/a -c}$
$\iff \quad \sqrt{a}x = -\frac{1}{2}b/\sqrt{a} + - \sqrt{\frac{1}{4}b^2/a -c} $
$\iff \quad x=-\frac{1}{2}b/a +- \sqrt{\frac{1}{4}b^2/a -c}/\sqrt{a}$
$\iff \quad x=-\frac{1}{2}b/a +- \sqrt{\frac{1}{4}b^2/a^2 -c/a}$
$\iff \quad \dots$
After some manipulations you will end up with the desired result.
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