Quadratic formula calculator
Calculator
The calculator needs JavaScript. The tables below give the same numbers for common cases.
How it is worked out
- Equation: 1x² − 3x + 2 = 0.
- Discriminant D = b² − 4ac = 9 − 8 = 1.
- D is positive: two real roots, x = (−b ± √D) ÷ 2a = 2 and 1.
| Inputs | Answer |
|---|---|
| a: 1; b: -3; c: 2 | 1x² − 3x + 2 = 0 has two real roots. |
| a: 1; b: 2; c: 5 | 1x² + 2x + 5 = 0 has two complex roots. |
| a: 2; b: 4; c: 2 | 2x² + 4x + 2 = 0 has one repeated real root. |
| a: -1; b: 0; c: 9 | -1x² + 0x + 9 = 0 has two real roots. |
How to use it
The solutions of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, D = b² − 4ac, is the discriminant: if D > 0 there are two real roots, if D = 0 one repeated root, and if D < 0 two complex roots.
The calculator also gives the vertex of the parabola, x = −b ÷ 2a, and whether it opens up (a > 0) or down. The coefficient a cannot be 0.
Frequently asked questions
What are the roots of x² − 3x + 2 = 0?
x = 1 and x = 2, since D = 9 − 8 = 1.
What does a negative discriminant mean?
The parabola does not cross the x-axis; the roots are a complex-conjugate pair.
How do I find the vertex?
x = −b ÷ 2a, then substitute it back to get y.
Method: standard arithmetic shown step by step; no outside data is used. Checked 2026-11-12.
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