It is often useful to know the roots of a quadratic function. We will first derive an expression for the inverse of a quadratic polynomial $y(x)$ and then find the roots by setting $y=0$. The general form of a quadratic polynomial is

We wish to solve for $x$. The most straightforward way to accomplish this is by completing the square. First we divide the equation by the coefficient of the quadratic term, $a$,

Now move all of the terms involving $x$ to the left side and move everything else to the right side of the equation.

The left hand side can be made into a perfect square by adding the square of half of the coefficient of the linear term

then the left can be written as

Now re-write the right side by finding a common denominator,

taking the square root of both sides,

Now isolate $x$ to get the inverse of a quadratic

The special case when $y=0$ yields the quadratic formula