Completing the square

Completing the square is a method for rewriting a quadratic expression in a variable such as $x$ using only one occurrence of the variable, by combining the $x^2$ and $x$ terms into a single square.

In general, the expression $x^2+bx+c$ can be rewritten as $(x+\frac{1}{2}b)^2-\frac{1}{4}b^2+c$, where the $(x+\frac{1}{2}b)^2$ term expands to give both the $x^2$ and $bx$ terms.

The expression $ax^2+bx+c$ can be rewritten similarly by first taking $a$ out as a factor, giving $ax^2+bx+c=a\bigl(x^2+\tfrac{b}{a}x+\tfrac{c}{a}\bigr),$ and then rewriting the quadratic expression within the parentheses as above.

This technique can be used to solve quadratic equations and to derive the quadratic formula.