What this calculator does
A quadratic equation describes a parabola, and solving it means finding where that parabola crosses the horizontal axis. It may cross twice, touch once, or miss entirely: and the discriminant tells you which before you calculate anything else.
This solver reports the roots in whichever form applies, including complex roots when the parabola never reaches the axis. It also gives the vertex, which is the turning point and the answer to most optimisation questions that quadratics are used for.
The formula
The quadratic formula comes from completing the square on the general form. The discriminant b² − 4ac is the part under the square root: positive gives two real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots. The vertex sits at x = −b/2a, midway between the roots when they are real.
| Term | Meaning |
|---|---|
| a, b, c | The coefficients in ax² + bx + c = 0. |
| Discriminant | b² − 4ac, which determines the nature of the roots. |
| Vertex | The turning point of the parabola. |
| Axis of symmetry | The vertical line through the vertex. |
The inputs explained
| Field | What to enter |
|---|---|
| a | The coefficient of x². If this is zero the equation is linear, and the calculator solves it as such. |
| b | The coefficient of x. |
| c | The constant term. |
When to use it
Solving for roots
The standard use: finding the values of x that satisfy the equation. Both roots are given, or the complex pair when there are no real solutions.
Finding a maximum or minimum
The vertex is the turning point. For a negative leading coefficient it is a maximum, for a positive one a minimum: which covers most profit-maximisation and projectile-height problems.
Checking your working
The sum and product of the roots equal −b/a and c/a respectively. Those two identities are a quick way to verify a hand-solved answer without redoing the whole thing.
Understanding when there is no solution
A negative discriminant means the parabola never reaches the axis. In a physical problem that usually means the scenario as posed cannot occur.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the constant term changes the roots
Fixing a = 1 and b = −3 while the constant term varies moves the parabola vertically.
| c | Roots | Discriminant | Vertex |
|---|---|---|---|
| -4 | x₁ = 4.000 x₂ = -1.000 | 25.000: two real roots | (1.500, -6.250) |
| 0 | x₁ = 3.000 x₂ = 0 | 9.000: two real roots | (1.500, -2.250) |
| 2 | x₁ = 2.000 x₂ = 1.000 | 1.000: two real roots | (1.500, -0.2500) |
| 2.25 | x = 1.500 (repeated) | 0: one real root | (1.500, 0) |
| 3 | 1.500 ± 0.8660i | -3.000: complex roots | (1.500, 0.7500) |
| 5 | 1.500 ± 1.658i | -11.000: complex roots | (1.500, 2.750) |
The effect of the leading coefficient
Changing a stretches the parabola and moves both the roots and the vertex.
| a | Roots | Discriminant | Vertex | Sum · product of roots |
|---|---|---|---|---|
| -2 | x₁ = -2.000 x₂ = 0.5000 | 25.000: two real roots | (-0.7500, 3.125) | -1.500 · -1.000 |
| -1 | x₁ = -3.562 x₂ = 0.5616 | 17.000: two real roots | (-1.500, 4.250) | -3.000 · -2.000 |
| 0.5 | x₁ = 5.236 x₂ = 0.7639 | 5.000: two real roots | (3.000, -2.500) | 6.000 · 4.000 |
| 1 | x₁ = 2.000 x₂ = 1.000 | 1.000: two real roots | (1.500, -0.2500) | 3.000 · 2.000 |
| 2 | 0.7500 ± 0.6614i | -7.000: complex roots | (0.7500, 0.8750) | 1.500 · 1.000 |
| 4 | 0.3750 ± 0.5995i | -23.000: complex roots | (0.3750, 1.438) | 0.7500 · 0.5000 |
Questions
What does the discriminant tell me?
Positive means two distinct real roots, zero means one repeated real root where the parabola touches the axis, and negative means two complex conjugate roots and no crossing. You can determine the nature of the solutions before solving.
What are complex roots?
Solutions involving the imaginary unit i, where i² = −1. They arise when the parabola never reaches the horizontal axis. They are perfectly valid mathematically and appear throughout engineering, particularly in signal processing and control theory.
What if a is zero?
Then it is not a quadratic but a linear equation, bx + c = 0, with a single root at −c/b. The calculator detects this and solves it accordingly.
How do I find the vertex without the formula?
The vertex sits at x = −b/2a, midway between the roots. Substitute that back into the equation to get the y-coordinate.
Can I factorise instead?
Often, and it is faster when the roots are simple integers. The formula always works, which is why it is the reliable fallback when factorisation is not obvious.
For general powers and roots, see the exponents calculator. For triangle geometry, try triangle from three sides.