The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots. A root is the value of the $x$ coordinate where the function crosses the $x$-axis. That is, it is the $x$-coordinate at which the function's value equals zero.

The discriminant for quadratic functions is:

$\Delta = b^2-4ac$ 

Where $a$, $b$, and $c$ are the coefficients in $f(x) = ax^2 + bx + c$. The number of roots of the function can be determined by the value of $\Delta$.

The Discriminant and the Quadratic Formula

Recall the quadratic formula:

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

where $a$, $b$ and $c$are the constants ($a$ must be non-zero) from a quadratic polynomial.  

The discriminant $\Delta =b^2-4ac$ is the portion of the quadratic formula under the square root.  

Positive Discriminant

If ${\Delta}$ is positive, the square root in the quadratic formula is positive, and the solutions do not involve imaginary numbers.

 $x={\dfrac{-b \pm \sqrt{\text{positive number}}}{2a}}$

Because adding and subtracting a positive number will result in different values, a positive discriminant results in two distinct solutions, and two distinct roots of the quadratic function.

Zero Discriminant

If ${\Delta}$ is equal to zero, the square root in the quadratic formula is zero:

$x={\dfrac{-b \pm \sqrt{0}}{2a}}$

Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.

Negative Discriminant

If ${\Delta}$ is less than zero, the value under the square root in the quadratic formula is negative:  

$x=\dfrac{-b \pm \sqrt{\text{negative number}}}{2a}$

This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.

Example

Consider the quadratic function:

$f(x)=x^2-x-2$

Using $1$ as the value of $a$, $-1$ as the value of $b$, and $-2$ as the value of $c$, the discriminant of this function can be determined as follows:

$\Delta =(-1)^2-4\cdot 1 \cdot (-2)$

$\Delta =9$

Because Δ is greater than zero, the function has two distinct, real roots. Checking graphically, we can confirm this is true; the zeros of the function can be found at $x=-1$ and $x=2$.

Example

Graph of a polynomial with the quadratic function $ f(x) = x^2 - x - 2$. Because the value is greater than 0, the function has two distinct, real zeros. The graph of shows that it clearly has two roots: the function crosses the $x$-axis at $x=-1$ and $x=2$.