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+\frac bax+\frac ca\right)`

Now complete the square:


`x^2+\frac bax+\frac ca=x^2+\frac bax+(b^2)/(4a^2)-(b^2)/(4a^2)+\frac ca=\left(x+\frac b{2a}\right)^2-(b^2)/(4a^2)+(4ac)/(4a^2)=\left(x+\frac b{2a}\right)^2-(b^2-4ac)/(4a^2)`

Now, redistributing the leading
`a` gives


`a\left(x+\frac b{2a}\right)^2-(b^2-4ac)/(4a)`