## Theorem

The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions:

$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c}} {2a}$

### Real Coefficients

Let $a, b, c \in \R$.

The quadratic equation $a x^2 + b x + c = 0$ has:

• Two real solutions if $b^2 - 4 a c > 0$
• One real solution if $b^2 - 4 a c = 0$
• Two complex solutions in $\C$ if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates.

## Proof

Let $a x^2 + b x + c = 0$. Then:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle 4 a^2 x^2 + 4 a b x + 4 a c$$ $$=$$ $$\displaystyle$$ $$\displaystyle 0$$ $$\displaystyle$$ $$\displaystyle$$ multiplying through by $4 a$ $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle \left({2 a x + b}\right)^2 - b^2 + 4 a c$$ $$=$$ $$\displaystyle$$ $$\displaystyle 0$$ $$\displaystyle$$ $$\displaystyle$$ completing the square $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle \left({2 a x + b}\right)^2$$ $$=$$ $$\displaystyle$$ $$\displaystyle b^2 - 4 a c$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle x$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac {-b \pm \sqrt {b^2 - 4 a c} }{2a}$$ $$\displaystyle$$ $$\displaystyle$$

$\blacksquare$

## Also known as

This result is often referred to as the quadratic formula.