Quadratic Equations

The quadratic equation $f(x)=x^2+px+q$ has zero, one or two roots. To see this we add a term to create a complete square:

$$x^2+px+q=0$$ (1)

$$x^2+px+\frac{1}{4}p^2 = \frac{1}{4}p^2 -q$$ (2)

Now we can apply the binomial equation:

$$\left( x+\frac{p}{2} \right)^2 = \frac{1}{4}p^2 -q$$ (3)

We take the square root on both sides of the equation

$$x+\frac{p}{2} = \sqrt{\frac{p^2}{4} -q}$$ (4)

to get

$$x_{1,2} = -\frac{p}{2} \pm \sqrt{\frac{p^2}{4} -q}$$ (5)

The expression in the root

$$D = {\frac{p^2}{4} -q}$$ (6)

is called ``discriminant''. The quadratic equation has two real roots for $D>0$ and none for $D<0$. If $D=0$, we have a double root.