Introduction: A function of the form $y(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are numbers ($a\neq 0$) is called a quadratic function.
Zeros: The zeros of a quadratic function $y(x)=ax^2+bx+c$ are determined by solving the quadratic equation
\[
ax^2+bx+c=0.
\]
A quadratic equation can be solved using the quadratic formula. In this formula the discriminant is a key component. The discriminant of a quadratic equation $ax^2+bx+c=0$ is equal to $b^2-4ac$ and is denoted by $D$,
\[
D=b^2-4ac.
\]
We obtain the following discriminant criterion for a quadratic equation.
Discriminant criterion
For a quadratic equation $ax^2+bx+c=0$ with $a\neq 0$, the following holds for $D=b^2-4ac$:
- if $D>0$, then the solutions of the quadratic equation are:\[
x=\frac{-b-\sqrt{b^2-4ac}}{2a}
\text{ and }
x=\frac{-b+\sqrt{b^2-4ac}}{2a}.
\] - if $D=0$, then the solution of the quadratic equation is: $$\begin{align} x & = \frac{-b}{2a}.
\end{align}$$
(Note that the two solutions for $D>0$ in this case coincide, such that there is a unique solution.) - if $D<0$, then the quadratic equation has no solutions.