Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots, with the discriminant, vertex and axis of symmetry.
The quadratic formula
Any equation of the form ax² + bx + c = 0, with a not equal to zero, can be solved with the quadratic formula:
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant tells you what to expect
The expression under the square root, D = b² − 4ac, is called the discriminant:
- D > 0: two different real roots. The parabola crosses the x-axis twice.
- D = 0: one repeated real root. The parabola touches the x-axis at its vertex.
- D < 0: no real roots, two complex roots. The parabola never meets the x-axis.
For x² − 3x − 10 = 0: D = 9 + 40 = 49, and √49 = 7. So x = (3 ± 7) ÷ 2, giving x = 5 and x = −2. Check: (x − 5)(x + 2) = x² − 3x − 10. ✓
Vertex and axis of symmetry
The graph of a quadratic is a parabola. Its turning point, the vertex, sits at x = −b ÷ 2a. Substituting that x back into the equation gives the y-coordinate. If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.
Other ways to solve
Factoring is fastest when the roots are whole numbers. Completing the square is how the quadratic formula is derived and is useful for finding the vertex form a(x − h)² + k. The formula always works, which is why it is the method used here.
Frequently asked questions
What if a equals zero?
Then the equation is linear (bx + c = 0), not quadratic, and its single solution is x = −c ÷ b.
What does a negative discriminant mean?
It means the equation has no real solutions. The roots are complex numbers of the form p ± qi, which the calculator shows.
How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a. Substitute it into the equation to find the y-coordinate.
Last reviewed Oct 20, 2026. Results are estimates; check important decisions with a qualified professional.