The Quadratic Formula

The general second-order (real) polynomial is

$$p(x) \isdef a x^2 + b x + c \protect$$ (2.1)

where the coefficients $a,b,c$ are any real numbers, and we assume $a\neq 0$ since otherwise it would not be second order. Some experiments plotting $p(x)$ for different values of the coefficients leads one to guess that the curve is always a scaled and translated parabola. The canonical parabola centered at $x=x_0$ is given by

$$y(x) = d\cdot (x-x_0)^2 + e \protect$$ (2.2)

where the magnitude of $d$ determines the width of the parabola, and $e$ provides an arbitrary vertical offset. If $d>0$, the parabola has the minimum value $e$ at $x=x_0$; when $d<0$, the parabola reaches a maximum at $x=x_0$ (also equal to $e$). If we can find $d,e,x_0$ in terms of $a,b,c$ for any quadratic polynomial, then we can easily factor the polynomial. This is called completing the square. Multiplying out the right-hand side of Eq. (2.2) above, we get

$$y(x) = d(x-x_0)^2 + e = d x^2 -2 d x_0 x + d x_0^2 + e. \protect$$ (2.3)

Equating coefficients of like powers of $x$ to the general second-order polynomial in Eq. (2.1) gives

$$\begin{eqnarray*} d &=& a\\ -2 d x_0 &=& b \quad\Rightarrow\quad x_0 = -b/(2a) \\ d x_0^2 + e &=& c \quad\Rightarrow\quad e = c - b^2/(4a). \end{eqnarray*}$$

Using these answers, any second-order polynomial $p(x) = a x^2 + b x + c$ can be rewritten as a scaled, translated parabola

$$p(x) = a\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right).$$

In this form, the roots are easily found by solving $p(x)=0$ to get

$$\zbox {x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.}$$

This is the general quadratic formula. It was obtained by simple algebraic manipulation of the original polynomial. There is only one ``catch.'' What happens when $b^2 - 4ac$ is negative? This introduces the square root of a negative number which we could insist ``does not exist.'' Alternatively, we could invent complex numbers to accommodate it.