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

The quadratic formula says that
x=b±b24ac2ax=\dfrac{-\goldD{b}\pm\sqrt{\goldD{b}^2-4\purpleD{a}\redD{c}}}{2\purpleD{a}}
for any quadratic equation like:
ax2+bx+c=0\purpleD{a}x^2 + \goldD{b}x + \redD{c} = 0

What is the discriminant?

The discriminant\goldD{\text{discriminant}} is the part of the quadratic formula under the square root.
x=b±b24ac2ax=\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:
6x2+10x1=06x^2+10x-1 =0
From the equation, we see:
  • a=6a=6
  • b=10b=10
  • c=1c=-1
Plugging these values into the discriminant, we get:
b24ac=1024(6)(1)=100+24=124\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 xx-axis at two points. In other words, there are two solutions that have a yy-value of 00, so there must be two solutions to our original equation: 6x2+10x1=06x^2+10x-1 =0.

Practice

Problem 1
f(x)=3x2+24x+48f(x) = 3x^2+24x+48
What is the value of the discriminant of ff?
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
How many distinct real number zeros does ff have?
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Want more practice? Check out this exercise.
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