Question

How to complete the square with a leading coefficient?

Answer 1

Factor out the leading coefficient:

$ax^2+bx+c=a\left(x^2+\dfrac bax+\dfrac ca\right)$

Now complete the square:

$x^2+\dfrac bax+\dfrac ca=x^2+\dfrac bax+\dfrac{b^2}{4a^2}-\dfrac{b^2}{4a^2}+\dfrac ca=\left(x+\dfrac b{2a}\right)^2-\dfrac{b^2}{4a^2}+\dfrac{4ac}{4a^2}=\left(x+\dfrac b{2a}\right)^2-\dfrac{b^2-4ac}{4a^2}$

Now, redistributing the leading $a$ gives

$a\left(x+\dfrac b{2a}\right)^2-\dfrac{b^2-4ac}{4a}$