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# Discriminant review

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, equals, start fraction, minus, start color #e07d10, b, end color #e07d10, plus minus, square root of, start color #e07d10, b, end color #e07d10, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #e84d39, c, end color #e84d39, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction
start color #7854ab, a, end color #7854ab, x, squared, plus, start color #e07d10, b, end color #e07d10, x, plus, start color #e84d39, c, end color #e84d39, equals, 0

## What is the discriminant?

The start color #e07d10, start text, d, i, s, c, r, i, m, i, n, a, n, t, end text, end color #e07d10 is the part of the quadratic formula under the square root.
x, equals, start fraction, minus, b, plus minus, square root of, start color #e07d10, b, squared, minus, 4, a, c, end color #e07d10, end square root, divided by, 2, a, end fraction
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:
6, x, squared, plus, 10, x, minus, 1, equals, 0
From the equation, we see:
• a, equals, 6
• b, equals, 10
• c, equals, minus, 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.
A coordinate plane. The x- and y-axes each scale by one. The graph is a parabola function that opens up. The function decreases through negative two, two and has an x-intercept around negative two. The function has a minimum around negative one, negative five, then it increases through zero, negative one and has another x-intercept around zero.
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: 6, x, squared, plus, 10, x, minus, 1, equals, 0.

## Practice

Problem 1
f, left parenthesis, x, right parenthesis, equals, 3, x, squared, plus, 24, x, plus, 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.