How to use
- Enter the coefficients a, b and c. Decimals and fractions such as 1/2 are fine.
- Read the roots, the discriminant, the vertex and the factored form.
- Follow the steps and check the graph, where the roots are the points where the curve meets the x-axis.
Options explained
- Discriminant Δ = b² − 4ac
- Positive — two different real roots; zero — one repeated root; negative — no real roots but two complex roots.
- Exact roots
- Square roots are simplified (√72 = 6√2) and common factors cancelled, so x = (−2 ± √12) / 2 is shown as −1 ± √3. Decimals and fractions in a, b and c are cleared first by multiplying the whole equation by the same number, which does not change the roots.
- Vertex and axis
- The vertex is (−b/2a, c − b²/4a), the turning point of the parabola; the axis of symmetry is the vertical line through it.
Examples
| Equation | Discriminant | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 1 | 3 and 2 |
| x² − 4x + 4 = 0 | 0 | 2 (repeated) |
| x² + 2x − 2 = 0 | 12 | −1 ± √3 |
| x² + 2x + 5 = 0 | −16 | −1 ± 2i |
| 2x² + 3x − 2 = 0 | 25 | 1/2 and −2 |
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. The ± gives the two roots: one with + and one with −.
Frequently asked questions
What if a = 0?
Then the equation is linear (bx + c = 0) and has one solution, x = −c/b. The solver handles that case too.
When can a quadratic be factored?
Over the rational numbers, exactly when the discriminant is a perfect square. The factored form is shown in that case, e.g. 2x² + 3x − 2 = (2x − 1)(x + 2).
Why are complex roots written with i?
i is the square root of −1. When Δ < 0 the roots are a ± bi, a pair of complex conjugates; the graph does not touch the x-axis.
Privacy
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