How to use
- Write your equation as ax² + bx + c = 0.
- Enter a, b and c. Use 0 for a missing term. Decimals (0.5) and fractions (2/3) are fine.
- Read the roots, the discriminant, the vertex and the step-by-step working.
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0:
x = (−b ± √(b² − 4ac)) / 2a
The part under the root, D = b² − 4ac, is the discriminant. It tells you what kind of roots to expect before you finish.
| Discriminant | Roots | Example |
|---|---|---|
| Positive perfect square | Two rational roots | 2x² + 9x − 5 = 0 |
| Positive, not a perfect square | Two irrational roots | 2x² + 10x + 11 = 0 |
| Zero | One repeated root | 4x² − 20x + 25 = 0 |
| Negative | Two complex roots | x² + x + 2 = 0 |
The graph of y = ax² + bx + c is a parabola. Its vertex is at x = −b/2a, which is also the axis of symmetry.
Examples
- 2x² + 9x − 5 = 0: D = 81 + 40 = 121 = 11², so x = (−9 ± 11)/4 = 1/2 or −5. Factored: (2x − 1)(x + 5) = 0.
- 2x² + 10x + 11 = 0: D = 100 − 88 = 12 and √12 = 2√3, so x = (−5 ± √3)/2, about −1.633974596 and −3.366025404.
- x² + x + 2 = 0: D = 1 − 8 = −7, so x = (−1 ± √7 i)/2, about −0.5 ± 1.322875656i.
- Vertex of y = x² + 4x + 3: x = −4/2 = −2 and y = 4 − 8 + 3 = −1, so the vertex is (−2, −1).
Fractions, accuracy and checking your answer
When a, b or c are fractions, the calculator first multiplies every term by the lowest common denominator. For example, ½x² + ⅔x − ⅓ = 0 becomes 3x² + 4x − 2 = 0. Exact answers then stay exact.
Decimal roots use a rearranged form of the formula that avoids losing digits when b² is much larger than 4ac.
To check roots, their sum is −b/a and their product is c/a.
Need to simplify a root by hand? The square root calculator shows each step, for example √12 = 2√3.