How to use this calculator
- Enter or choose a (coefficient of x²).
- Enter or choose b (coefficient of x).
- Enter or choose c (constant).
- The solutions updates as you type. Use Copy result or Share to keep it.
Method
x = (−b ± √(b² − 4ac)) ÷ 2a; Discriminant D = b² − 4ac; Vertex = (−b ÷ 2a, c − b² ÷ 4a)
- a, b, c
- the coefficients, with a ≠ 0
- D
- discriminant: > 0 two real roots, = 0 one repeated root, < 0 two complex roots
Worked example
- D = (−5)² − 4 × 1 × 6 = 25 − 24 = 1.
- x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, so x = 3 or x = 2.
- Check: (x − 3)(x − 2) = x² − 5x + 6.
x² − 5x + 6 = 0 has two real solutions, x = 3 and x = 2.
What the result means
The real roots are where the parabola y = ax² + bx + c crosses the x-axis. Complex roots mean it never crosses; the vertex is then the closest it gets.
Assumptions and limits
- Coefficients are real numbers and a is not 0.
- Roots are shown to 10 significant digits. The calculator uses the numerically stable form of the formula, so a tiny root next to a huge one is still accurate.
Questions
What if a is 0?
Then there is no x² term and the equation is linear: bx + c = 0 gives x = −c ÷ b.
What does i mean in the answer?
i is the imaginary unit, √−1. A negative discriminant gives two complex roots p + qi and p − qi, which are conjugates of each other.