Quadratic Equation Solver
Solve ax² + bx + c = 0 with full step-by-step working, discriminant, and vertex information.
How to use
- Enter the three coefficients a, b, and c for the equation ax² + bx + c = 0.
- The equation preview updates live as you type so you can verify your input.
- a must not be zero — otherwise the equation is linear, not quadratic.
- Click Solve to get the roots, discriminant, vertex, and axis of symmetry.
- When the discriminant is negative, the roots are complex (a ± bi format).
- Full step-by-step working using the quadratic formula is shown below the results.
About this Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are known numbers and a cannot be zero (otherwise it isn't quadratic). Solving it means finding the value(s) of x that make the equation true — usually two solutions, sometimes one, sometimes none in real numbers.
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant tells you what kind of answer to expect
The expression under the square root (b² − 4ac) is called the discriminant. If it's positive, there are two distinct real solutions. If it equals zero, there is exactly one real solution (a repeated root). If it's negative, there are no real solutions — only two complex (imaginary) solutions, since you can't take the square root of a negative number within real numbers.
Worked example
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): the discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, a positive number, so there are two real roots. x = (5 ± 1) ÷ 2, giving x = 3 or x = 2. Checking: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 — correct.