Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots, with the discriminant, vertex, axis of symmetry, step list and a parabola plot. How it works
Use the Quadratic Equation Solver
About the Quadratic Equation Solver
x² − 3x + 2 = 0 has the roots 2 and 1: the discriminant b² − 4ac is 1, which is above 0, so there are two real roots. Its parabola has the vertex (1.5, -0.25) and the axis of symmetry x = 1.5, so its vertex form is y = (x − 1.5000)² − 0.2500. This quadratic equation solver works the same out for any a, b and c, giving complex roots when the discriminant is below 0.
Quadratics solved by the tool
| a, b, c | Discriminant | Roots | Vertex | Factorized form |
|---|---|---|---|---|
| 1, -3, 2 | 1.0000 | 2.0000, 1.0000 | (1.5000, -0.2500) | (x - 2)(x - 1) |
| 2, -7, 3 | 25.0000 | 3.0000, 0.5000 | (1.7500, -3.1250) | 2(x - 3)(x - 1/2) |
| 1, 0, -4 | 16.0000 | -2.0000, 2.0000 | (0.0000, -4.0000) | (x + 2)(x - 2) |
| 1, -4, 4 | 0.0000 | 2.0000, 2.0000 | (2.0000, 0.0000) | (x - 2)² |
| 3, -4, 1 | 4.0000 | 1.0000, 0.3333 | (0.6667, -0.3333) | 3(x - 1)(x - 1/3) |
| 1, 0, -2 | 8.0000 | -1.4142, 1.4142 | (0.0000, -2.0000) | none |
| 1, 2, 5 | -16.0000 | -1.0000 + 2.0000i, -1.0000 - 2.0000i | (-1.0000, 4.0000) | none |
Values are shown to 4 decimals, as on the page. "none" means the tool shows no factorized form because the roots are not both exact fractions.
How to use this tool
- Type a, b and c from ax² + bx + c = 0 into the three fields, which start at 1, -3 and 2. Enter 0 for a missing b or c term. The coefficient a cannot be 0.
- Read the discriminant and the root type: two distinct real roots, one repeated real root or two complex conjugate roots.
- Read the roots, the vertex and the axis of symmetry. The factorized form appears when both roots are exact fractions, and the step list, the plot and the function in standard, vertex and factored form follow.
Frequently asked questions
- How do you solve a quadratic equation with the quadratic formula?
- Work out the discriminant D = b² − 4ac, then x = (−b ± √D) / 2a. For 2x² − 7x + 3 = 0, D = 49 − 24 = 25 and x = (7 ± 5) / 4, so x is 3 or 0.5. The solver does this for any a that is not 0 and writes the steps out.
- What does the discriminant tell you about the roots?
- Its sign decides the root type. Above 0 there are two real roots: x² − 4 = 0 has D = 16 and the roots -2 and 2. At 0 there is one repeated root: x² + 2x + 1 = 0 has D = 0 and the root -1. Below 0 the roots are complex conjugates: x² + 2x + 5 = 0 has D = -16 and the roots -1 + 2i and -1 - 2i.
- How do you find the vertex of a parabola from a, b and c?
- The vertex is at h = −b / 2a and k = −D / 4a. For x² − 3x + 2, h = 1.5 and k = -0.25, so the vertex is (1.5, -0.25) and the axis of symmetry is x = 1.5. The vertex is the lowest point of the parabola when a is positive and the highest when a is negative. The tool also writes the vertex form, y = (x − 1.5000)² − 0.2500.
- Why is there no factorized form for x² − 2 = 0?
- Its roots, -1.4142 and 1.4142, are not fractions, so there is no factorization with exact numbers. The tool shows a factorized form only when both roots are exact fractions with a denominator of at most 10,000, as in 3x² − 4x + 1 = 0, which factorizes as 3(x - 1)(x - 1/3). The other-forms panel still writes y = (x + 1.4142)(x − 1.4142) for x² − 2, marked as rounded to 4 decimals. Complex roots get no factored form.
- What if a is 0?
- Then the equation is a straight line, not a quadratic, and the tool says so instead of solving it. With a = 0 and b = 0 as well, it says there is no solution (c is not 0) or that the equation is an identity (c is 0).
- How large or small can the coefficients be?
- Large ones work until b², 4ac or a root passes about 1.8 × 10^308, the largest number a browser can hold, and then the tool shows a message. It also refuses coefficients that are all smaller than 1 × 10^-100, where b² and 4ac would be lost to rounding: multiply the whole equation by a common factor and the roots stay the same.
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