Quadratic Formula Solver
Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, plus the discriminant that tells you which.
How it works
x = (−b ± √(b² − 4ac)) ÷ (2a) · discriminant Δ = b² − 4ac
The quadratic formula solves any equation ax² + bx + c = 0 in one step. The discriminant b² − 4ac decides the nature of the roots: positive gives two distinct real roots, zero gives one repeated root where the parabola is tangent to the x-axis, and negative gives a conjugate pair of complex roots. This solver checks the discriminant first and reports the correct case.
Worked example
For x² − 3x + 2 = 0, a = 1, b = −3, c = 2. The discriminant is (−3)² − 4·1·2 = 9 − 8 = 1 > 0, so there are two real roots: x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2, giving x = 2 and x = 1.
Sources: Quadratic formula — Wikipedia
Frequently asked questions
What does the discriminant tell me?
The discriminant Δ = b² − 4ac is the part under the square root. Δ > 0 means two different real roots, Δ = 0 means one repeated real root, and Δ < 0 means no real roots (two complex conjugate roots instead). You can read off the root type without finishing the calculation.
What happens if a is zero?
If a = 0 the x² term disappears and it is no longer a quadratic — it becomes the linear equation bx + c = 0, whose single solution is x = −c ÷ b. The solver detects this and returns the linear root instead of dividing by zero.
How are complex roots written here?
When the discriminant is negative the roots come as a conjugate pair p ± qi, where p = −b ÷ 2a is the real part and q = √(−Δ) ÷ 2a is the imaginary part. The two roots are mirror images across the real axis.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.