Completing the Square

Theorem
Let $a, b, c, x$ be real numbers with $a \ne 0$.

Then:


 * $a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$

This process is known as completing the square.

Also presented as
This result can also be presented in the form:


 * $a x^2 + b x + c = \dfrac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}$

Also see

 * Tschirnhaus Transformation
 * Quadratic Formula