About the Quadratic Formula Calculator
A quadratic formula calculator solves equations of the form ax² + bx + c = 0, returning both roots and identifying whether they are real or complex. Quadratics describe projectile paths, optimisation problems, and any relationship where a squared term dominates.
The formula
x = (−b ± √(b² − 4ac)) / 2aa, b, and c are the coefficients, with a non-zero. The expression b² − 4ac under the root is the discriminant, and its sign determines the nature of the solutions.
How to use this calculator
- 1Enter your Coefficient a. The field starts at
1, which you can overwrite. - 2Enter your Coefficient b. The field starts at
-5, which you can overwrite. - 3Enter your Coefficient c. The field starts at
6, which you can overwrite. - 4Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Coefficient a | 1 |
| Coefficient b | -5 |
| Coefficient c | 6 |
Result
Discriminant: 1.0000
x₁ = 3.000000
x₂ = 2.000000
Vertex: (2.5000, -0.2500)
Sum of Roots: 5.0000
Product of Roots: 6.0000
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
The discriminant tells you the answer's character before you compute it. Positive means two distinct real roots, so the parabola crosses the x-axis twice. Zero means one repeated root, where the parabola just touches the axis. Negative means no real roots and two complex conjugates, so the curve never reaches the axis at all.
Geometrically the formula finds where a parabola meets the x-axis. The vertex sits at x = −b/2a, exactly midway between the roots, and the ± in the formula is the symmetric step either side of it. That symmetry is why the sum of the roots is always −b/a and their product always c/a.
Things worth knowing
- If a is zero the equation is linear, not quadratic, and this formula does not apply.
- Check the discriminant first — it tells you immediately whether to expect real solutions.
- Try factoring before using the formula. Many quadratics with integer roots factor quickly.
- The vertex at x = −b/2a gives the maximum or minimum, which is what optimisation problems need.
- Roots sum to −b/a and multiply to c/a, which is a fast way to check your answers.
Frequently asked questions
What is the quadratic formula?+
x = (−b ± √(b² − 4ac)) / 2a, which solves any equation of the form ax² + bx + c = 0. The ± produces the two roots.
What does the discriminant tell me?+
The value of b² − 4ac determines the roots. Positive gives two distinct real roots, zero gives one repeated root, and negative gives two complex conjugate roots with no real solution.
What does it mean when there are no real solutions?+
That the parabola never crosses the x-axis. The solutions exist as complex numbers involving i, which matters in electrical engineering and physics but means no real-world crossing point.
Should I factor instead of using the formula?+
Factoring is faster when the roots are simple integers. The formula always works, so it is the reliable fallback when factoring is not obvious.