Quadratic Equation Solver
Find real or complex roots of ax² + bx + c = 0 with the discriminant, quadratic formula and factoring shown step by step.
Press Enter to solve · ^ for powers · sqrt() for roots · separate a system with a comma
This quadratic equation solver works as a quadratic formula calculator and a factoring tool in one. Enter any equation of degree two and it identifies a, b and c, calculates the discriminant, factors when possible and returns both roots in exact and decimal form.
How to use the quadratic equation solver
- Type the equation, e.g.
2x^2 - 5x - 3 = 0. It does not need to be in standard form. - Press Enter.
- Read the discriminant, the formula with your numbers substituted and the vertex of the parabola.
The quadratic formula
The expression under the square root, Δ = b² − 4ac, is the discriminant. It tells you how many real solutions to expect before you finish the calculation:
- Δ > 0 — two different real roots
- Δ = 0 — one repeated real root
- Δ < 0 — two complex roots, no x-intercepts
Factoring vs. the quadratic formula
Factoring is fastest when the roots are whole numbers or simple fractions. Completing the square shows the vertex form. The quadratic formula works for every quadratic, so it is the safest choice when factoring is not obvious.
Example: solve 2x² − 5x − 3 = 0
- a = 2, b = −5, c = −3.
- Δ = (−5)² − 4(2)(−3) = 25 + 24 = 49.
- x = (5 ± 7) / 4.
- x = 3 or x = −1/2.
Common mistakes
- Using +b instead of −b at the start of the formula.
- Dividing only the square root by 2a instead of the whole numerator.
- Stopping when Δ is negative — the roots are complex, not “no answer”.
Questions about quadratic equations
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree two that can be written as ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola.
How many solutions does a quadratic equation have?
Exactly two solutions counted with multiplicity: two real roots, one repeated real root, or two complex conjugate roots, depending on the discriminant.
Can this calculator give complex roots?
Yes. When the discriminant is negative, the solver returns both complex roots in a + bi form, for example −2 ± 2i.