Calculators
Quadratic Equation Solver
Solve quadratic equations and inspect roots, discriminant, and vertex.
Quadratic Equation Solver tool
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About This Tool
Solve a quadratic equation in standard form ax² + bx + c = 0 by entering its three coefficients. The solver handles two real roots, a repeated real root, and complex conjugate roots. It also reports the discriminant, axis of symmetry, and vertex, connecting the equation's algebraic solutions with the shape of its parabola. Calculations run locally in your browser and accept integer, decimal, and negative coefficients.
How To Use It
- Rewrite the equation in standard form ax² + bx + c = 0.
- Enter the coefficients a, b, and c, including negative signs where needed. The value of a cannot be zero.
- Read x₁ and x₂ for the roots, then use the discriminant and vertex information to interpret the quadratic.
Examples
Two real roots
For x² − 5x + 6 = 0, enter a = 1, b = −5, c = 6. The discriminant is 1 and the roots are 2 and 3.
Repeated root
For x² + 2x + 1 = 0, the discriminant is 0, so both roots are x = −1. The parabola touches the x-axis at its vertex.
Complex roots
For x² + 1 = 0, enter a = 1, b = 0, c = 1. The discriminant is −4 and the roots are i and −i, displayed as complex conjugates.
Decimal coefficients
Coefficients do not have to be whole numbers. For −0.5x² + 1.5x + 1 = 0, enter the decimals directly and the same quadratic formula applies.
Useful Notes
Quadratic formula
The roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / (2a). The quantity inside the square root is the discriminant, usually written Δ = b² − 4ac.
What the discriminant tells you
When Δ > 0 there are two distinct real roots. When Δ = 0 there is one repeated real root. When Δ < 0 the equation has two complex conjugate roots rather than real x-intercepts.
Axis and vertex
The parabola's axis of symmetry is x = −b/(2a). Substituting that x-value into ax² + bx + c gives the vertex y-coordinate. If a is positive the vertex is a minimum; if a is negative it is a maximum.
Numerical precision
Results use JavaScript floating-point arithmetic and are displayed to a practical precision. Very large coefficients that overflow reliable numeric calculation are rejected rather than returning misleading infinite values.
FAQ
Why can't a equal zero?
If a = 0, the x² term disappears and the equation becomes linear rather than quadratic. A linear equation should be solved with its simpler linear formula.
Can a quadratic equation have no real roots?
Yes. A negative discriminant means there are no real x-intercepts, but there are still two solutions in the complex numbers.
What does a repeated root mean?
When the discriminant is zero, both quadratic-formula branches give the same x-value. Graphically, the parabola touches the x-axis once at that point.
Does the order of the two roots matter?
No. x₁ and x₂ label the two solutions; swapping their display order does not change the solution set.