Extra explanation: zeros

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.