# A calculus of the absurd

The quadratic formula 1414 This is the result of completing the square for $$ax^2 + bx + c$$ and using it solve the equation $$ax^2 + bx + c = 0$$ for $$x$$ states that for a quadratic $$ax^2 + bx + c = 0$$,

$$\label {the quadratic formula} x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}$$

The $$b^2 - 4ac$$ gives quite a bit of information away. It’s called the "discriminant", and sometimes written as $$\Delta$$.

If $$b^2 - 4ac > 0$$ then we have exactly two solutions to the quadratic.

If $$b^2 - 4ac = 0$$ (which also means that $$\sqrt {b^2 - 4ac} = 0$$), then there will only be one solution (because adding zero does nothing 1515 This is often handy for algebraic manipulation. - see the equation directly below for details).

\begin{equation*} \frac {-b \pm \sqrt {b^2 - 4ac}}{2a} = \frac {-b}{2a} \text { if $b^2 - 4ac = 0$} \end{equation*}

If $$b^2 - 4ac < 0$$ then there are no real solutions to the quadratic.