Discriminant of a Quadratic Equation - Definition & Examples

The Discriminant

When does $ax^2+bx+c=0$ have a real solution? Well, we know how to find the solutions using the quadratic formula:

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

That makes it look like we can always find solutions. However, we can't always take a square root! Sometimes you'll get a negative number under the radical. A negative under the radical means there are no real number solutions to the radical. We can use the formula under the radical, $b^2-4ac$, called the discriminant, to determine the number of roots of solutions in a quadratic equation. There are three cases:

• $b^2-4ac < 0$: The equation has $0$ real solutions. The graph does not cross the $x$-axis.
• $b^2-4ac = 0$: The equation has $1$ real solution. The graph touches the $x$-axis at one point.
• $b^2-4ac > 0$: The equation has $2$ real solutions. The graph crosses through the $x$-axis at two points.

To learn more about how to determine solutions using the x-intercepts, check out this lesson. Using the discriminant will be really helpful when you're trying to quickly solve quadratic equation word problems later on!

Discriminant

Let's do a little review before we investigate the discriminant. Given a quadratic equation in standard form, $$ax^2+bx+c=0,$$ where $a, b,$ and $c$ are constants and $a\neq 0,$ we can use the quadratic formula, $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a},$$ to find the solutions of the equation.

The discriminant of a quadratic equation is $b^2-4ac,$ the quantity under the square root sign in the quadratic formula. $$x=\frac{-b\pm \sqrt{\boxed{b^2-4ac}}}{2a}$$ $$\text{Discriminant}=\boxed{b^2-4ac}$$

If the discriminant is greater than $0,$ the quadratic equation has $2$ real solutions.

If the discriminant is equal to $0,$ the quadratic equation has $1$ real solution.

If the discriminant is less than $0,$ the quadratic equation has $0$ real solutions. (Instead of real solutions, the quadratic equation has $2$ imaginary solutions.)

When the discriminant is negative, the quadratic formula produces complex roots. If this happens, you can also try factoring with complex numbers. We'll dive into these concepts later!

(Video) The Discriminant

by Eddie Woo

This video by Eddie Woo explains the discriminant.

$\large{\text{Summary}}$

The discriminant is the part of the quadratic formula under the radical. It is used to tell you how many roots you will have. $$x=\frac{-b\pm\sqrt{\color{#8925AE}{b^2-4ac}}}{2a}\\ \text{Discriminant: }b^2-4ac$$

In general, there are three possible outcomes for the discriminant:

1. $b^2-4ac<0\rightarrow$ No real roots. Roots will have imaginary numbers. This means the quadratic is either always above the axis, or always below.
2. $b^2-4ac=0\rightarrow$ One real root. The graph touches the $x$-axis in one place.
3. $b^2-4ac>0\rightarrow$ Two real roots. The graph crosses the $x$-axis twice.