Quadratic Equations in the Complex Number System

Quadratic Equations in the Complex Number System

For the general quadratic equation $ax^2+bx+c=0$ with $a,b,c\in\mathbb{R}$ and $a\neq0$, completing the square gives

the familiar quadratic formula, where $D=b^2-4ac$ is the discriminant. Until now, this formula broke down whenever $D<0$, since $\sqrt{D}$ had no real value. With $i$ available, it no longer breaks down at all: writing the negative discriminant as $D=-|D|$ (where $|D|>0$),

so the quadratic formula gives the two roots

These two roots are complex conjugates of each other (same real part $-b/2a$, imaginary parts equal in magnitude but opposite in sign) — a direct consequence of the $\pm$ in the formula, and true precisely because $a,b,c$ are real. Every quadratic equation with real coefficients is therefore now solvable in $\mathbb{C}$, with no exceptions: real-and-distinct roots when $D>0$, one repeated real root when $D=0$, and a genuine conjugate pair of complex roots when $D<0$.

$x^2+2x+5=0$: $a=1,b=2,c=5$, so $D=4-20=-16$, $\sqrt{D}=\sqrt{-16}=4i$. The roots are $x=\dfrac{-2\pm4i}{2}=-1\pm2i$, i.e. $x=-1+2i$ and $x=-1-2i$ — a conjugate pair, as expected.

Sum and product of the roots. Whatever the sign of $D$, if $\alpha,\beta$ are the two roots produced by the formula, direct computation from $x=\dfrac{-b\pm\sqrt D}{2a}$ gives

exactly as for real roots — these standard relations (sometimes called Vieta's formulas) continue to hold unchanged in the complex number system. …