Quadratic Equation Solver

Solve ax^2 + bx + c = 0. Shows discriminant, real or complex roots, vertex, and parabola graph.

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1. Enter the coefficients a, b, and c for the equation ax^2 + bx + c = 0. 2. View the discriminant value and what it tells you about the nature of the roots. 3. See both roots displayed - real or complex, depending on the discriminant. 4. Review the vertex coordinates and axis of symmetry for the parabola. 5. Examine the interactive parabola graph showing roots and vertex. 6. Click the copy button to copy the roots or any other value.

About This Tool

The Quadratic Equation Solver finds the roots of any quadratic equation in the standard form ax^2 + bx + c = 0. Simply enter the coefficients a, b, and c, and the tool instantly computes the discriminant, determines whether the roots are real and distinct, real and repeated, or complex conjugates, and displays the exact solutions in simplified form.

Beyond finding roots, the solver provides a complete analysis of the corresponding parabola. It calculates the vertex coordinates, axis of symmetry, direction of opening (upward or downward), and y-intercept. A dynamic graph plots the parabola with clearly marked roots, vertex, and axis of symmetry so you can visualize the equation and verify the solutions at a glance.

This tool is essential for algebra and pre-calculus students who need to check their work or understand how changing coefficients affects the shape and position of a parabola. It is equally useful for engineers and scientists who encounter quadratic relationships in physics, optimization problems, and signal processing. Every step is shown so you can follow the logic from discriminant to final answer.

Formula / How It Works

x = (-b +/- sqrt(b^2 - 4ac)) / 2a | Discriminant: D = b^2 - 4ac | Vertex: (-b/2a, f(-b/2a))

Frequently Asked Questions

The discriminant is D = b^2 - 4ac. It tells you the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one repeated real root; if D < 0, the roots are complex conjugates with imaginary parts.
Yes. When the discriminant is negative, the solver computes and displays the complex conjugate roots in the form p + qi and p - qi, where i is the imaginary unit.
The vertex is the highest or lowest point on the parabola. Its x-coordinate is -b/(2a) and the y-coordinate is found by plugging that value back into the equation. The vertex represents the maximum value when a < 0 or the minimum value when a > 0.
Yes. A dynamic graph plots the parabola with the roots, vertex, and axis of symmetry clearly marked. The graph automatically scales to show all important features of the curve.
If a = 0, the equation becomes linear (bx + c = 0) rather than quadratic. The solver detects this and either solves the linear equation or prompts you to enter a non-zero value for a.

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