Completing the Squre

It changes the form of a quadratic equation. $ax^2+bx+c=0 \mspace{15mu} \Longrightarrow \mspace{15mu} a(x+d)^2+e=0$

Shortcuts

$d=\frac{b}{2a}$ $e=c-\frac{b^2}{ra}$

Method

To solve $ax^2+bx+c=0$ by completing the square:

1. Transform so that the constant, $c$, is alone on the right side. $ax^2+bx=-c$
2. If the leading coefficient, $a$, is not equal to $1$, divide both sides by $a$. $x^2+\frac{b}{a}x=\frac{-c}{a}$
3. Add the square of half the coefficient of the $x$-term, $(\frac{b}{2a})^2$, to both sides.
4. Factor the left side as the square of the binomial. (The inside of square should be $x-\frac{b}{2a}$).