The calculator solves a quadratic equation ax² + bx + c = 0. When the discriminant is negative, the roots are written as a complex pair a + bi. The steps use the same numbers that appear in the result.
Coefficients
The equation is ax² + bx + c = 0.
Lesson: when a quadratic root is complex
A quadratic equation always has two roots in the complex numbers, counting multiplicity. Over the real numbers the picture depends on the discriminant Δ = b² − 4ac.
A positive discriminant gives two distinct real roots. A zero discriminant gives one real root (a repeated root). A negative discriminant means there is no real number whose square equals Δ, so the roots leave the real line. They appear as a conjugate pair: the same real part and opposite imaginary parts.
Coefficient a cannot be 0 on this page. That would collapse the equation to a linear equation, which this calculator does not treat. For real roots only, the quadratic formula calculator stops when Δ is negative and points back here.
Practice problems
1. Negative discriminant
Solve x² + 2x + 5 = 0.
2. Positive discriminant
Solve 2x² − 3x − 2 = 0.
3. Repeated root
Solve x² − 6x + 9 = 0.
Show answers
Problem 1. Δ = −16. The roots are −1 + 2i and −1 − 2i.