Quadratic Equation Solver
Solve any equation of the form ax² + bx + c = 0. You get both roots — real or complex — along with the discriminant, the vertex, the axis of symmetry and the factored form where one exists, with the working shown at each step.
Solve a quadratic equation
How to use this calculator
- Write your equation in the form ax² + bx + c = 0, moving everything to one side if it is not already.
- Enter a, b and c. Include the sign: in x² − 3x + 2 = 0, b is −3.
- Press Calculate for the roots, the discriminant, the vertex and the axis of symmetry, with the working shown.
What the discriminant tells you
The discriminant is b² − 4ac, the part of the quadratic formula under the square root. Before solving anything it tells you what kind of answer to expect:
- Positive — two distinct real roots. The parabola crosses the x-axis twice.
- Zero — one repeated real root. The parabola touches the x-axis at its vertex.
- Negative — no real roots. The parabola never meets the x-axis, and the two roots are complex conjugates. This calculator gives them rather than stopping at "no solution".
Why this does not always use the plain quadratic formula
Applying the formula literally loses accuracy when b² is much larger than 4ac. One of the two roots is then computed by subtracting two nearly equal numbers, and most of the significant digits cancel — a well known trap called catastrophic cancellation.
This calculator avoids it: it computes the root with the numerically stable sign first, then finds the other from the fact that the product of the roots equals c/a. The answers are the same in exact arithmetic and considerably more accurate in floating point.
Frequently Asked Questions
How do I solve a quadratic equation using the quadratic formula?
Write the equation as ax squared plus bx plus c equals zero, then apply x equals minus b plus or minus the square root of b squared minus 4ac, all over 2a. Enter a, b and c above and the calculator shows each step, including the discriminant.
What does the discriminant tell me?
The discriminant is b squared minus 4ac. If it is positive there are two distinct real roots; if it is zero there is one repeated root; if it is negative there are no real roots and the two solutions are complex conjugates.
Can this solve equations with no real solutions?
Yes. When the discriminant is negative the calculator reports the two complex roots in the form p plus or minus qi, rather than simply saying there is no solution.
Why can't a be zero?
With a equal to zero there is no x squared term, so the equation is linear rather than quadratic and the quadratic formula divides by zero. Solve bx plus c equals zero directly instead: x equals minus c over b.
What are the vertex and the axis of symmetry?
The vertex is the turning point of the parabola, at x equals minus b over 2a. The axis of symmetry is the vertical line through it. The vertex is a minimum when a is positive and a maximum when a is negative.