Quadratic Formula Calculator
Solve quadratic equations and find real or complex roots with step-by-step working.
Enter the coefficients for the equation ax² + bx + c = 0. The coefficient a must not be zero.
Roots
Discriminant (D = b² − 4ac)
Vertex & Axis of Symmetry
Step-by-step
The Quadratic Formula
Any quadratic equation written as ax² + bx + c = 0 (with a ≠ 0) can be solved using the quadratic formula:
x = ( −b ± √(b² − 4ac) ) / (2a)
The ± symbol means there are generally two solutions: one using addition and one using subtraction of the square root term.
What the Discriminant Tells You
The discriminant is the part under the square root, D = b² − 4ac. Its sign determines the nature of the roots:
- D > 0: two distinct real roots.
- D = 0: one repeated real root.
- D < 0: two complex conjugate roots of the form p ± q i.
The Vertex Form
Every quadratic describes a parabola. Its turning point, the vertex, sits at x = −b/(2a), and its y-value is c − b²/(4a). The vertical line through the vertex, x = −b/(2a), is the axis of symmetry: the parabola is a mirror image on either side of it.
Tip: If a > 0 the parabola opens upward and the vertex is a minimum; if a < 0 it opens downward and the vertex is a maximum.
Related Calculators
Note: This calculator solves quadratic equations using the quadratic formula. Results are rounded for display and are intended for educational and informational purposes. While we strive for accuracy, please verify important calculations independently.