Quadratic Equation Solver
Quadratic Equation Solver
Enter the coefficients a, b, and c from your quadratic equation in standard form (ax² + bx + c = 0). This calculates the discriminant and both roots using the quadratic formula, handling real, repeated, and complex root cases automatically.
The quadratic formula works for every quadratic equation, unlike factoring which only works cleanly when the roots happen to be simple whole or rational numbers — which is why it's the go-to general-purpose method. The discriminant (the part under the square root) determines what kind of roots to expect before you even finish the calculation: positive gives two distinct real roots, exactly zero gives one repeated real root, and negative gives a pair of complex roots, which this calculator reports correctly rather than showing an error.
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a — the discriminant (b² − 4ac) under the square root determines what kind of roots the equation has.
- Discriminant sign determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
- Coefficient a can't be zero: if a = 0, the equation is linear, not quadratic — this solver assumes a genuine quadratic (a ≠ 0).
What does it mean when the roots are complex?
It means the parabola described by the equation never crosses the x-axis — there's no real number that makes the equation equal zero, so the solutions are expressed with the imaginary unit i instead.
Why do some quadratics have only one root instead of two?
When the discriminant equals exactly zero, the two roots from the ± in the formula collapse to the same value — this happens when the parabola just touches the x-axis at a single point (its vertex) rather than crossing it twice.
Quadratic Equation Solver
Enter the coefficients a, b, and c from your quadratic equation in standard form (ax² + bx + c = 0). This calculates the discriminant and both roots using the quadratic formula, handling real, repeated, and complex root cases automatically.
The quadratic formula works for every quadratic equation, unlike factoring which only works cleanly when the roots happen to be simple whole or rational numbers — which is why it's the go-to general-purpose method. The discriminant (the part under the square root) determines what kind of roots to expect before you even finish the calculation: positive gives two distinct real roots, exactly zero gives one repeated real root, and negative gives a pair of complex roots, which this calculator reports correctly rather than showing an error.
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a — the discriminant (b² − 4ac) under the square root determines what kind of roots the equation has.
- Discriminant sign determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
- Coefficient a can't be zero: if a = 0, the equation is linear, not quadratic — this solver assumes a genuine quadratic (a ≠ 0).
What does it mean when the roots are complex?
It means the parabola described by the equation never crosses the x-axis — there's no real number that makes the equation equal zero, so the solutions are expressed with the imaginary unit i instead.
Why do some quadratics have only one root instead of two?
When the discriminant equals exactly zero, the two roots from the ± in the formula collapse to the same value — this happens when the parabola just touches the x-axis at a single point (its vertex) rather than crossing it twice.