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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.

## Want to join the conversation?

• "A discriminant of zero indicates that the quadratic has a repeated real number solution." what exactly does this mean?
• Or in factored form, it would look like (x+3)^2 or (x-3)^2, 3 is just an example number, it could be anything. On a graph, the parabola's vertex would be on the x axis, so it only intersects the x axis at that one point.
• I don't understand what F(x) means? The f symbol just appeared
(1 vote)
• f(x) is read as f of x, and it means a function in terms of x. This is called functional notation, and it has the same meaning as y = at this point in math. As you get into Algebra II, you will learn how to combine functions where this language will be more useful than the y = form of equations. The biggest use of f(x) in Algebra I is when you are asked to find a specific value of x. So if f(x) = 2x + 6, this is equivalent to y = 2x+ 6, but if I wanted to find the value of the function at x = 8, with functional notation, I could just say f(8) which is solved by putting 8 into x and getting f(8) = 22.
• if the eqaution has no real roots , use the discriminant to determine the value of n.
0=5.5x^2+nx+n and the discriminant is -40.

This is another homework question I dont know how to do this.
• basically you're looking b and c, which in this case are the same, so you can plug everything into the discriminant equation (b^2 -4ac):
n^2 -4(5.5)(n)=-40
i don't know if i'm being dumb and there's an easier way to solve this but you can simplify this to:
n^2 -11n +40 =0
which, you'll notice, is a quadratic equation, so you just solve for that to get n.
(1 vote)
• how can the discriminant help graph?
• It determines the number of times the graph crosses the x-axis.
Discriminant > 0: the graph crosses the x-axis twice
Discriminant = 0: the graph touches the x-axis at its maximum or minimum point
Discriminant < 0: The graph has no x-intercepts, which means it is wholly above or below the x-axis
• How do you find the discriminant from looking at a graph?
• I don't think there's an easy way to find the exact value of the discriminant by looking at the graph, but looking at the graph can tell you if the discriminant is positive, negative, or zero.

If the graph doesn't touch the x axis at all, the discriminant is negative
If the graph touches the x axis a only one point, the discriminant is zero
If the graph touches the x axis at two distinct points, the discriminant is positive.

Sorry I couldn't give you an easy answer, but if you know the equation, then it's pretty easy to find the discriminant, so I don't know if it's worth it to learn how to find it from only the graph.
• Why do we need the discriminant? We already know what kind of solutions there are when we solve using the quadratic formula.
• The quadratic equation discriminant is important because it tells us the number and type of solutions.
• How is a quadratic equation with a negative discriminant graphed?
• You just don't have x-intercepts to work with.
You can graph it using a table of values -- pick values for X and calculate Y for each X.
You can still find the vertex and axis of symmetry.
• I have a question that was given to me in class, it is:
x^2 - (k+4)x + k + 7 = 0. Find k.
Answer: k = -6 & 2

I understand HOW to put this into the discriminant and get the correct answer, but not WHY we do that. How come we have to use the discriminant to find k? How do I know when I need to use this for equations?
• The answers that you found (for k) are when the discriminant equal 0 (b^2-4ac=0) -- which means that the function has only one solution.
When you graph (k+4)^2-4(k+7), you get a convex parabola with vertex (-2,-16) and x-intercepts at (-6,0) and (2,0).
That implies that for k; -6<k<2, that the discriminant is negative. In other words there is no real solution for those values of k.
For k=-6 & k=2, which you found the function (with x) has only one x-intercept (which is the vertex).
For k<-6 & k>2, the function has two solutions (x-intercepts).
So, you find the discriminant in order to figure out for which values for k, the function has 0, 1 or 2 solutions.