Math
Discriminant Calculator
Enter a, b, and c and the discriminant calculator computes b^2 - 4ac exactly, tells you how many real roots the quadratic has and what kind they are, checks whether D is a perfect square (so the quadratic factors), and previews the roots in exact form.
Cannot be 0.
Enter 0 if there is no x term.
Enter 0 if there is no constant.
Try an example
Result
Discriminant D = b^2 - 4ac
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- Nature of the roots
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- Number of real roots
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- Perfect square?
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- √|D| simplified
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- Roots
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- Factors over the rationals?
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More details (2 more)
- x-intercepts of the graph
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- Vertex
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Study path
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What the discriminant calculator solves
The The b² − 4ac part of the quadratic formula; its sign tells you how many real solutions exist. is the part of the quadratic formula under the square root. Its sign tells you, before solving, whether ax^2 + bx + c = 0 has two real solutions, one repeated solution, or two complex solutions. Whether it is a perfect square tells you whether the quadratic factors nicely.
Classification table
| Discriminant | Number of real roots | Type of roots | Example |
|---|---|---|---|
| D > 0 and a perfect square | 2 | Rational, distinct; quadratic factors | x^2 - 3x + 2: D = 1 |
| D > 0, not a perfect square | 2 | Irrational, distinct (contain a radical) | x^2 - 3x - 2: D = 17 |
| D = 0 | 1 (repeated) | Rational double root; perfect-square trinomial | x^2 - 6x + 9: D = 0 |
| D < 0 | 0 | Two complex conjugates a ± bi | x^2 + 2x + 5: D = -16 |
How to use it
- Write the equation as ax^2 + bx + c = 0 and read off a, b, and c with their signs.
- Enter the three numbers. The discriminant appears as an exact value with a decimal when needed.
- Use the classification row to answer 'how many real roots' questions, and the perfect-square row to decide whether to factor or use the formula.
How to read the answer
A positive discriminant means the graph crosses the x-axis twice, zero means it touches once at the The turning point of a parabola, or a corner point of a shape., and negative means it never touches. A perfect-square discriminant means the roots are fractions or whole numbers.
Common mistakes and edge cases
- Squaring b incorrectly: (-3)^2 is 9, not -9.
- Dropping the sign of c: for x^2 - 3x - 2, c = -2, so -4ac = +8.
- Treating D = 0 as 'no roots'. It is one real root counted twice.
- Using a = 0. A linear equation has no discriminant.
Worked examples
Two rational roots
x^2 - 3x + 2: D = 1
Discriminant D = b^2 - 4ac
1
Two irrational roots
x^2 - 3x - 2: D = 17
Discriminant D = b^2 - 4ac
17
Repeated root
x^2 - 6x + 9: D = 0
Discriminant D = b^2 - 4ac
0
Complex roots
x^2 + 2x + 5: D = -16
Discriminant D = b^2 - 4ac
-16
Decimal coefficients
0.5x^2 + 1.5x - 1: D = 4.25 = 17/4
Discriminant D = b^2 - 4ac
17/4 (4.25)
a = 0 is not quadratic
0x^2 + 2x + 1
Discriminant D = b^2 - 4ac
Error
Frequently asked questions
What is the discriminant?+
The discriminant of ax^2 + bx + c is b^2 - 4ac. It is the expression under the square root in the quadratic formula, and its sign decides what kind of roots the equation has.
What does a negative discriminant mean?+
The square root of a negative number is not real, so the equation has no real solutions. The two solutions are complex conjugates of the form a ± bi, and the parabola never crosses the x-axis.
What does a discriminant of 0 mean?+
The ± in the quadratic formula adds and subtracts 0, so both solutions are the same number: one repeated (double) root at x = -b/(2a). The vertex of the parabola sits on the x-axis.
How does the discriminant tell me if a quadratic factors?+
If D is a perfect square (0, 1, 4, 9, 16, ...), √D is a whole number and the roots are rational, so the quadratic factors over the integers or rationals. If D is positive but not a perfect square, the roots contain a radical and it does not factor nicely.
Can the discriminant be used for cubic equations?+
Cubics have their own discriminant with a different formula (18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2). Use the cubic equation calculator for that.
Why does the calculator show a fraction for the discriminant?+
When a, b, or c are decimals, the exact discriminant is often a fraction such as 17/4. The decimal (4.25) is shown next to it.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026