The Quadratic Formula

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

\displaystyle y = ax^2 + bx + c

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,

\displaystyle \frac{y}{a} = x^2 + \frac{b}{a}x + \frac{c}{a}

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

\displaystyle x^2 + \frac{b}{a}x = \frac{y}{a} - \frac{c}{a}

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

\displaystyle x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2= \left(\frac{b}{2a}\right)^2 + \frac{y}{a} - \frac{c}{a}

then the left can be written as

\displaystyle \left(x + \frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 + \frac{y}{a} - \frac{c}{a}

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

\displaystyle \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 + 4ay - 4ac}{4a^2}

\displaystyle \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 + 4a(y - c)}{4a^2}

taking the square root of both sides,

\displaystyle x + \frac{b}{2a} = \frac{\pm\sqrt{b^2 + 4a(y - c)}}{2a}

Now isolate x to get the inverse of a quadratic

\displaystyle x = \frac{-b\pm\sqrt{b^2 + 4a(y - c)}}{2a}

The special case when y=0 yields the quadratic formula

\displaystyle x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}