Math
Quadratic Formula Calculator
Enter a, b, and c, or type the equation itself, and the quadratic formula calculator gives both roots in exact form (simplified radicals, reduced fractions, or a ± bi), the decimal values, the discriminant, vertex, axis of symmetry, factored form, vertex form, and a labeled graph with every step of the work.
Cannot be 0. Decimals and negatives are fine; the calculator converts them to exact fractions.
Use 0 if there is no x term.
Use 0 if there is no constant.
Try an example
Result
Roots (exact)
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- Roots (decimal)
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- Discriminant
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- Type of roots
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- Vertex
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- Factored form
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- Vertex form
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the quadratic formula solves
Any equation that can be written as ax^2 + bx + c = 0 with a ≠ 0 is a quadratic. The quadratic formula finds every solution (root) directly from the three The number multiplying a variable, like the 3 in 3x., even when the quadratic does not factor. This page also shows what the roots mean on the graph: they are the x-A point where a graph crosses an axis. of the parabola y = ax^2 + bx + c.
How to use this calculator
- Choose Coefficients and enter a, b, c from ax^2 + bx + c = 0, or choose Type the equation and enter it as written, for example x^2 = 4x - 3.
- Read the exact roots at the top of the result. A number that cannot be written as a fraction of two whole numbers, like π or √2. roots stay as simplified A root expression written with the √ symbol, such as a square root. such as (3 ± √17)/2; rational roots are reduced fractions.
- Check the The b² − 4ac part of the quadratic formula; its sign tells you how many real solutions exist. to see why the roots are real, repeated, or complex.
- Open Show the work to copy the substitution, discriminant, radical simplification, and both roots line by line.
How to read the discriminant
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0, perfect square | Two different rational roots | Crosses the x-axis twice; factors over the integers or rationals |
| D > 0, not a perfect square | Two different irrational roots | Crosses the x-axis twice; roots contain a radical |
| D = 0 | One repeated real root | Vertex sits on the x-axis |
| D < 0 | Two complex conjugate roots a ± bi | Never touches the x-axis |
Exact form vs decimal form
Teachers usually want the exact answer. √12 simplifies to 2√3, and (4 ± √12)/2 reduces to 2 ± √3 because 2 divides every term. The calculator does that simplification for you and then shows the decimal approximation separately so you can check a graph or a calculator answer.
Common mistakes
- Forgetting that -b flips the sign of b: if b = -3, then -b = 3.
- Squaring a negative b incorrectly: (-3)^2 = 9, not -9.
- Dividing only the radical by 2a. The whole The top number of a fraction; it counts how many parts you have. -b ± √D is divided by 2a.
- Leaving the equation in a form such as x^2 = 4x - 3. Move everything to one side first, or use Type the equation mode and let the calculator do it.
- Trying to use a = 0. That is a linear equation, not a quadratic, and 2a in the The bottom number of a fraction; it says how many equal parts make one whole. would be 0.
Worked examples
Two irrational roots
x^2 - 3x - 2 = 0 gives (3 ± √17)/2
Roots (exact)
x = (3 ± √17)/2
Two rational roots
2x^2 + 5x - 3 = 0 factors as (x + 3)(2x - 1)
Roots (exact)
x = -3 or x = 1/2
Repeated root
x^2 - 6x + 9 = 0 has the double root x = 3
Roots (exact)
x = 3 (double root)
Complex roots
x^2 + 2x + 5 = 0 gives -1 ± 2i
Roots (exact)
x = -1 ± 2i
Decimal coefficients
0.5x^2 - 1.5x + 1 = 0 is solved exactly as x^2 - 3x + 2 = 0
Roots (exact)
x = 1 or x = 2
Typed equation
x^2 = 4x - 3 is rearranged to x^2 - 4x + 3 = 0
Roots (exact)
x = 1 or x = 3
Typed with unicode and simplified radical
x² − 4x + 1 = 0 gives 2 ± √3 after simplifying √12
Roots (exact)
x = 2 ± √3
a = 0 is not a quadratic
0x^2 + 2x - 4 = 0 is linear
Roots (exact)
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Frequently asked questions
What is the quadratic formula?+
For ax^2 + bx + c = 0, the roots are x = (-b ± √(b^2 - 4ac)) / (2a). The ± gives the two solutions: one with plus, one with minus.
What does the discriminant tell me?+
The discriminant b^2 - 4ac decides the type of roots. Positive means two real roots (rational if it is a perfect square), zero means one repeated root, and negative means two complex roots.
How do I write the roots in exact form?+
Keep the square root instead of a decimal, simplify it (√12 = 2√3), and divide every term in the numerator by any common factor with 2a. This calculator shows that simplified form automatically.
Can the quadratic formula give complex answers?+
Yes. When b^2 - 4ac is negative, √(negative) becomes i times a real radical, so the roots look like a ± bi. They are still valid solutions, just not real numbers.
Why does my textbook answer look different from the calculator's?+
Usually it is the same value in a different form: 2 ± √3 versus (4 ± √12)/2, or 1/2 versus 0.5. Compare the decimal values to confirm.
Can I type the equation instead of entering a, b, c?+
Yes. Switch to Type the equation and enter it however it appears, such as x^2 = 4x - 3 or 3x² + 2x − 5 = 0. The calculator moves everything to one side and reads off a, b, and c.
What if a = 0?+
Then the equation is linear, not quadratic, and the formula breaks because 2a = 0. Solve bx + c = 0 directly: x = -c/b.
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