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Discriminant Calculator

Enter a, b and c to get the discriminant and what it says about the roots.

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of x squared
of x
constant
Try:

Discriminant

Δ=64\Delta = 64Δ=64

Two real roots, x = 3 and x = -1.

Forms of the same parabola

Standard formy=2x2−4x−6y = 2x^{2} - 4x - 6y=2x2−4x−6
Factored formy=2(x−3)(x+1)y = 2\left(x - 3\right)\left(x + 1\right)y=2(x−3)(x+1)
Vertex formy=2(x−1)2−8y = 2\left(x - 1\right)^2 - 8y=2(x−1)2−8
Vertex(1, −8)\left(1,\ -8\right)(1, −8)
Axis of symmetryx=1x = 1x=1
y-intercept(0, −6)\left(0,\ -6\right)(0, −6)
DiscriminantΔ=64\Delta = 64Δ=64

Factors over the rational numbers. The parabola opens upward, so the vertex is a minimum.

Parabola

  • y
Table of values
Graph of the parabola with its roots and vertex marked: sampled values across the visible range
xy
-3.838.08
-2.8421.491
-1.888.5888
-0.92-0.6272
0.04-6.1568
1-8
1.96-6.1568
2.92-0.6272
3.888.5888
4.8421.491
5.838.08

Quadratic formula, step by step

  1. 1Identify the coefficients
    ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
    ↓
    2x2−4x−6=02x^{2} - 4x - 6 = 02x2−4x−6=0

    a = 2, b = -4, c = -6

  2. 2Clear fractions
    2x2−4x−6=02x^{2} - 4x - 6 = 02x2−4x−6=0
    ↓
    x2−2x−3=0x^{2} - 2x - 3 = 0x2−2x−3=0

    Multiplying the whole equation by the same number does not change its roots.

  3. 3Discriminant
    Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac
    ↓
    Δ=(−2)2−4(1)(−3)=16\Delta = \left(-2\right)^2 - 4(1)\left(-3\right) = 16Δ=(−2)2−4(1)(−3)=16

    Positive: two different real roots.

  4. 4Quadratic formula
    x=−b±Δ2ax = \frac{-b \pm \sqrt{\Delta}}{2a}x=2a−b±Δ​​
    ↓
    x=2±162x = \frac{2 \pm \sqrt{16}}{2}x=22±16​​
  5. 5Square root of the discriminant
    16\sqrt{16}16​
    ↓
    444

    16 is a perfect square.

  6. 6The two roots
    x=2±42x = \frac{2 \pm 4}{2}x=22±4​
    ↓
    x1=3,x2=−1x_1 = 3, \quad x_2 = -1x1​=3,x2​=−1

Steps

  1. 1Divide by a
    2x2−4x−6=02x^{2} - 4x - 6 = 02x2−4x−6=0
    ↓
    x2−2x=3x^2 - 2x = 3x2−2x=3

    Divide every term by a so the x squared term has coefficient 1.

  2. 2Add the square of half the x coefficient
    (b2a)2\left(\frac{b}{2a}\right)^2(2ab​)2
    ↓
    (−1)2=1\left(-1\right)^2 = 1(−1)2=1

    Add this to both sides.

  3. 3Write the left side as a square
    x2−2x+1=4x^2 - 2x + 1 = 4x2−2x+1=4
    ↓
    (x−1)2=4\left(x - 1\right)^2 = 4(x−1)2=4

    The right side equals the discriminant over 4a squared: 64 / 16.

  4. 4Take the square root of both sides
    x−1=±4x - 1 = \pm\sqrt{4}x−1=±4​
    ↓
    x−1=±2x - 1 = \pm 2x−1=±2
  5. 5Vertex form
    y=2x2−4x−6y = 2x^{2} - 4x - 6y=2x2−4x−6
    ↓
    y=2(x−1)2−8y = 2\left(x - 1\right)^2 - 8y=2(x−1)2−8

    The same completing-the-square work, kept as an expression, gives the vertex.

  1. Tools
  2. Mathematics
  3. Quadratic Formula Calculator
  4. Discriminant Calculator

About Discriminant Calculator

This discriminant calculator finds b² - 4ac for any quadratic and tells you at once whether the equation has two real roots, one repeated root or a complex pair. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The discriminant is the part of the quadratic formula under the square root. You can read the type of roots from its sign without solving the whole equation, which is why it is the first thing teachers ask for in classification questions.

The page opens on 2x² - 4x - 6 = 0, whose discriminant is 64. Because 64 is a perfect square, the roots are rational, and the calculator also shows them with the full formula working, the vertex and the graph.

How to use Discriminant Calculator

  1. 1Choose what to show: Roots, Discriminant, Vertex or Completing the square.
  2. 2Enter a, b and c in their boxes, or switch to Equation and type something like 2x^2 - 7x + 3 = 0.
  3. 3Read the headline answer. It updates as you type, with no calculate button.
  4. 4Open the steps underneath to follow the substitution into the quadratic formula, or the completing-the-square route.
  5. 5Use the table for the factored form, vertex form, axis of symmetry and y-intercept, and the graph to see where the parabola crosses the x axis.

Features

  • Exact roots: rational, simplified radical or complex, never only a rounded decimal.
  • Discriminant with a reading: two real roots, one repeated root or a complex pair.
  • Vertex, axis of symmetry, y-intercept and whether the parabola opens up or down.
  • Factored form over the rationals when it exists, with the complex factorisation otherwise.
  • Completing the square written out, ending in vertex form.
  • Quadratic formula steps in KaTeX, with a copy button for the LaTeX.
  • Interactive plot with roots and vertex marked, zoom, pan and PNG export.
  • Accepts integers, decimals, fractions and a full equation.

The discriminant

Compute b squared minus four times a times c.

Δ = b² - 4ac

Worked examples

  • 2x² - 4x - 6 = 0 → gives Δ = 64

    Positive and a perfect square, so the roots are rational: 3 and -1.

  • x² + 2x + 5 = 0 → gives Δ = -16

    Negative, so there are no real roots and the parabola stays above the x axis.

Privacy: how it works

  • Everything is calculated by code running in your browser tab. The coefficients you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Check an answer by substituting a root back into the equation: the left side should come out as 0.
  • If the roots look awkward, look at the discriminant first. A perfect square means rational roots and a clean factorisation.
  • The sum of the roots is -b/a and their product is c/a, which is a quick check on any pair of roots.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Coefficients are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Coefficients are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What is the discriminant of a quadratic?

It is the number b² - 4ac. It decides how many real roots the equation has: two when positive, one when zero and none when negative.

Why is the discriminant useful without solving?

Its sign gives the type of roots, and a positive perfect square also tells you the quadratic factors over the integers.

More Quadratic Formula Calculator pages

  • Quadratic Formula CalculatorEnter a, b and c, or type the equation, and get the roots with every step, the vertex and a graph of the parabola.
  • Vertex Form CalculatorEnter a, b and c to get the vertex, the axis of symmetry and the vertex form.
  • Completing the Square CalculatorEnter a, b and c to see the equation rewritten as a perfect square, step by step.
  • Quadratic Equation With Complex RootsSolve a quadratic with no real roots and get the complex pair, with the working.

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