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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

The quadratic formula says that
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
for any quadratic equation like:
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:
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.
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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
How many distinct real number zeros does f have?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want more practice? Check out this exercise.

Want to join the conversation?

  • purple pi purple style avatar for user Shuss824
    "A discriminant of zero indicates that the quadratic has a repeated real number solution." what exactly does this mean?
    (17 votes)
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  • aqualine sapling style avatar for user Kathy Downey
    I don't understand what F(x) means? The f symbol just appeared
    (1 vote)
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    • mr pink green style avatar for user David Severin
      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.
      (32 votes)
  • aqualine seed style avatar for user Anirudh Parmar
    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.
    (5 votes)
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    • blobby green style avatar for user rylan.wetsell
      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)
  • duskpin seedling style avatar for user westina_7
    how can the discriminant help graph?
    (2 votes)
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    • aqualine tree style avatar for user Isabella C
      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
      (14 votes)
  • sneak peak green style avatar for user Sage
    How do you find the discriminant from looking at a graph?
    (2 votes)
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    • aqualine ultimate style avatar for user Hannah Alisse
      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.
      (13 votes)
  • blobby green style avatar for user drossington
    Why do we need the discriminant? We already know what kind of solutions there are when we solve using the quadratic formula.
    (6 votes)
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  • piceratops seed style avatar for user sunix777
    How is a quadratic equation with a negative discriminant graphed?
    (2 votes)
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  • duskpin sapling style avatar for user Bree
    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?
    (3 votes)
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    • starky ultimate style avatar for user Timo
      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.
      (6 votes)
  • starky sapling style avatar for user Rowaid Karabsheh
    I dont understand the graph!
    (4 votes)
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  • aqualine tree style avatar for user centc3231
    how do u tell if the discriamnt is postive, negitive, or zero on a graph
    (2 votes)
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    • starky sapling style avatar for user Sanya Singh
      If the graph has two x-intercepts, then it has a positive discriminant. If the graph has zero x-intercepts, then it has a negative discriminant. If the graph has only one x-intercept, then the discriminant is zero. Hope this helped you!!😊
      (7 votes)