Quadratic Formula Calculator
Solve ax² + bx + c = 0 and see every step, including the discriminant, the vertex and complex roots.
Results
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Steps
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Quick Answer
The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x + 2 = 0, the discriminant is 1, so x = 2 or x = 1. A positive discriminant gives two real roots, zero gives one, and a negative value gives two complex roots.
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Table of Contents
How This Quadratic Formula Calculator Works
A quadratic equation has the form ax² + bx + c = 0. Enter a, b and c, and the calculator finds the roots, the discriminant and the vertex. It also shows each step, so you can follow the working.
Quick explanation: the discriminant, b² − 4ac, tells you what type of roots to expect.
- Positive: two different real roots.
- Zero: one repeated real root.
- Negative: two complex roots.
Core Formulas
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a
- Discriminant: D = b² − 4ac
- Vertex x: −b ÷ 2a
- Vertex y: c − b² ÷ 4a
Worked Example
Solve x² + 2x + 5 = 0, so a = 1, b = 2 and c = 5.
- Discriminant = 2² − 4(1)(5) = −16
- √−16 = 4i
- x = (−2 ± 4i) ÷ 2
- x = −1 + 2i or x = −1 − 2i
Common Mistakes
- Equation not equal to zero: move all terms to one side first.
- Negative signs: if b is negative, −b becomes positive.
- Squaring a negative: (−3)² is 9, not −9.
- Dividing only part: divide the whole top by 2a.
Frequently Asked Questions
It is x = (−b ± √(b² − 4ac)) ÷ 2a, used to solve any equation of the form ax² + bx + c = 0 where a is not zero.
It shows the type of roots: positive gives two real roots, zero gives one, and negative gives two complex roots.
If a is 0, the x² term disappears, so the equation is linear and not quadratic.
Yes. If the discriminant is negative, it shows the two roots in a + bi form.
It is the highest or lowest point of the graph of y = ax² + bx + c. Its x value is −b ÷ 2a.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Quadratic formula | The formula and its derivation. | https://en.wikipedia.org/wiki/Quadratic_formula |
| Wikipedia, Discriminant | How the discriminant describes the roots. | https://en.wikipedia.org/wiki/Discriminant |