How this calculator works
Any equation of the form ax² + bx + c = 0 (with a ≠ 0) can be solved with the quadratic formula. The expression under the square root, b² − 4ac, is the discriminant: if it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are a complex-conjugate pair.
The calculator also gives the exact answer in simplest radical form when a, b and c are whole numbers, and the vertex of the parabola y = ax² + bx + c, which sits halfway between the two roots.
Assumptions and tips
- Decimal answers are rounded to 6 decimal places; the exact form is not rounded.
- Write the equation in standard form (everything on one side, = 0) before reading off a, b and c.
Worked example
Using the default values in the calculator above:
| Input | Value |
|---|---|
| a (x² coefficient) | 1 |
| b (x coefficient) | -3 |
| c (constant) | -10 |
| Result | Value |
|---|---|
| x₁ | 5 |
| x₂ | -2 |
| Roots | Two real roots |
| Exact form | x = 5 or x = -2 |
| Discriminant (b² − 4ac) | 49 |
| Vertex (h, k) | (1.5, -12.25) |
| Axis of symmetry | x = 1.5 |
| Parabola opens | Upward (minimum at the vertex) |
Step by step:
- Identify a = 1, b = -3, c = -10
- Discriminant D = b² − 4ac = 9 − 4 × 1 × -10 = 49
- D > 0: two real roots. √D = 7
- x = (3 ± 7) ÷ 2
- x₁ = 5, x₂ = -2
- Vertex: h = −b ÷ 2a = 1.5, k = c − b² ÷ 4a = -12.25
Worked examples
Each equation solved with this calculator.
| Equation | x₁ | Discriminant |
|---|---|---|
| x² − 3x − 10 = 0 | 5 | 49 |
| x² − 5x + 6 = 0 | 3 | 1 |
| 2x² + 3x − 2 = 0 | 0.5 | 25 |
| x² − 2 = 0 | 1.414214 | 8 |
| x² + 2x − 1 = 0 | 0.414214 | 8 |
| 3x² − 4x − 4 = 0 | 2 | 64 |
| x² − 7x + 12 = 0 | 4 | 1 |
| 4x² + 4x − 3 = 0 | 0.5 | 64 |
Common mistakes to avoid
- Dropping the sign of b: −b means change its sign, so b = −3 gives +3.
- Dividing only the square root by 2a — the whole numerator, −b ± √D, is divided by 2a.
- Reading a, b and c before moving every term to one side of the equation.
Frequently asked questions
How do I solve x² − 3x − 10 = 0?
a = 1, b = −3, c = −10. D = 9 + 40 = 49, √49 = 7, so x = (3 ± 7) / 2: x = 5 or x = −2.
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there are no real solutions — the two roots are complex numbers a ± bi.
What if a = 0?
Then the equation is linear, bx + c = 0, with the single solution x = −c / b. The calculator handles that case too.
How do I find the vertex of a parabola?
h = −b / (2a), and k is the value of the expression at x = h. For x² − 3x − 10 the vertex is (1.5, −12.25).
Sources
Built and reviewed by Anirban Kundu, Creator & Editor · M.Sc Physics, 20 years teaching science