Completing the Square

Theorem
Let $R$ be a commutative ring with unity.

Let $a,b,c,x \in R$.

Let $2a$ be invertible.

Then


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

This process is known as completing the square.

Also see

 * Tschirnhaus Transformation
 * Quadratic Formula