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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=b±b24ac2a
for any quadratic equation like:
ax2+bx+c=0

What is the discriminant?

The discriminant is the part of the quadratic formula under the square root.
x=b±b24ac2a
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=0
From the equation, we see:
  • a=6
  • b=10
  • c=1
Plugging these values into the discriminant, we get:
b24ac=1024(6)(1)=100+24=124
This is a positive number, so the quadratic has two solutions.
This makes sense if we think about the corresponding graph of y=6x2+10x1:
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.
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: 6x2+10x1=0.

Practice

Problem 1
f(x)=3x2+24x+48
What is the value of the discriminant of f?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
How many distinct real number zeros does f have?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want more practice? Check out this exercise.

Want to join the conversation?

  • 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.
    (7 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 ⇒ 𝑥 = (-𝑏 ± √(𝑏² – 4𝑎𝑐))/(2𝑎)

      Using this formula, it is advisable to calculate the discriminant, 𝑏² – 4𝑎𝑐, first because if it is negative we know that there are no real solutions and we can skip the rest of the calculations.
      (84 votes)
  • 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?
    (18 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.
      (37 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?
    (3 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
      (15 votes)
  • sneak peak green style avatar for user Sage
    How do you find the discriminant from looking at a graph?
    (3 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.
      (14 votes)
  • piceratops seed style avatar for user sunix777
    How is a quadratic equation with a negative discriminant graphed?
    (2 votes)
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  • blobby green style avatar for user Radha Krishna
    how discriminant decides what are the nature of the two roots?
    I mean how?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      The quadratic formula: x = [-B +/- sqrt(B^2-4AC)] / (2A)
      The discriminant is B^2-4AC. Notice this is the portion of the formula inside the square root.

      If the discriminant = 0, then the formula degrades to x = -B/(2A). So, there is only one solution.

      If the discriminant is positive, then the square root creates a real number. So, there are 2 real solutions.

      If the discriminant is negative, then the square root is not a real number. Square roots of negative values require the using of complex numbers. So, there are 2 solutions, that are not real numbers. Or, 2 complex solutions.

      Hope this helps.
      (13 votes)
  • 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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