# Discriminant review

CCSS Math: HSA.REI.B.4b
The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.

### Quick review of the quadratic formula

$x=\dfrac{-\goldD{b}\pm\sqrt{\goldD{b}^2-4\purpleD{a}\redD{c}}}{2\purpleD{a}}$
$\purpleD{a}x^2 + \goldD{b}x + \redD{c} = 0$

## What is the discriminant?

The $\goldD{\text{discriminant}}$ is the part of the quadratic formula under the square root.
$x=\dfrac{-{b}\pm\sqrt{\goldD{b^2-4ac}}}{2a}$
The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.
• A positive discriminant indicates that the quadratic has two distinct real number solutions.
• A discriminant of zero indicates that the quadratic has a repeated real number solution.
• A negative discriminant indicates that neither of the solutions are real numbers.
Want to understand these rules at a deeper level? Check out this video.

### Example

We're given a quadratic equation and asked how many solutions it has:
$6x^2+10x-1 =0$
From the equation, we see:
• $a=6$
• $b=10$
• $c=-1$
Plugging these values into the discriminant, we get:
\begin{aligned} &b^2-4ac\\\\ =&10^2-4(6)(-1)\\\\ =&100+24\\\\ =&124 \end{aligned}
This is a positive number, so the quadratic has two solutions.
This makes sense if we think about the corresponding graph.
Graph of y=6x^2+10x-1
Notice how it crosses the $x$-axis at two points. In other words, there are two solutions that have a $y$-value of $0$, so there must be two solutions to our original equation: $6x^2+10x-1 =0$.

## Practice

Problem 1
$f(x) = 3x^2+24x+48$
What is the value of the discriminant of $f$?
How many distinct real number zeros does $f$ have?

Want more practice? Check out this exercise.