Quadratic Formula Calculator: Roots, Discriminant & Parabola Graph
Free quadratic formula calculator: solve ax² + bx + c = 0. Find roots in simplified radical form and decimal form. Discriminant D = b² − 4ac, vertex (h,k), parabola with axis of symmetry. Complex roots when D < 0. How to solve quadratic equations; quadratic equation solver for algebra and math. Trusted by students and educators. No sign-up.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:Roots x₁ and x₂ in radical and decimal form; D = b² − 4ac; vertex (h, k) and y-intercept (0, c); a parabola visualizer with vertex, axis of symmetry, roots (or “no real x-intercepts” when D < 0), and y-intercept when in view; a logic trace (identify a, b, c; compute D; substitute into x = (−b ± √D)/(2a)). When D < 0, roots use the imaginary unit i (e.g. 1 ± 2i).
- Ideal Users:Students & algebra: Solve quadratic equations, get radical form and decimals, check homework. Teachers: Demonstrate the formula, discriminant, and parabola geometry. Anyone: Quick quadratic equation solver with clear steps and visuals. How to solve a quadratic equation: enter a, b, c and read the roots and graph.
- Scope & Limits:Real coefficients only. Coefficient a must be non-zero. All calculations run locally; no sign-up.
This quadratic formula calculator solves ax² + bx + c = 0 and is widely used in high school algebra and college math. It shows roots in simplified radical form (e.g. 2 ± 3√5) and decimal form, the discriminant, vertex, and a parabola graph with axis of symmetry and labeled points.
The Quadratic Formula and Discriminant
For ax² + bx + c = 0 with a ≠ 0, the roots are
The discriminant is
If D > 0, there are two distinct real roots. If D = 0, one repeated (double) root. If D < 0, the roots are complex conjugates: x = (−b ± i√|D|) / (2a). This quadratic formula calculator shows all three cases with simplified radical form (√D simplified, e.g. 2√5) and decimal approximations.
Vertex and Axis of Symmetry of the Parabola
The parabola f(x) = ax² + bx + c has vertex (h, k) where h = −b/(2a) and k = f(h). The axis of symmetry is the vertical line x = h. The calculator displays the vertex and draws the axis on the graph. When D < 0, the parabola does not cross the x-axis (no real roots); the graph shows the curve and a note that roots are complex.
How to Solve a Quadratic Equation Step by Step
To solve a quadratic equation using the formula: (1) Write the equation as ax² + bx + c = 0 and identify a, b, and c. (2) Compute D = b² − 4ac. (3) Substitute into x = (−b ± √D) / (2a); if D < 0, use √D = i√|D|. (4) Simplify radicals and decimals. This quadratic equation solver does steps 2–4 for you and shows the parabola, vertex, and logic trace.
Quadratic Formula Calculator FAQ
What is the quadratic formula?
For ax² + bx + c = 0 (a ≠ 0), the roots are
The calculator shows roots in simplified radical form and decimal form, and uses i when the discriminant is negative
What does the discriminant tell you?
The discriminant is
D > 0: two distinct real roots. D = 0: one repeated real root. D < 0: two complex conjugate roots. The tool displays D and the correct root form
How do I get simplified radical form?
When D ≥ 0, the calculator factors perfect squares out of √D (e.g. √20 → 2√5) and writes each root as a real part ± a multiple of √(radicand). Decimal approximations are shown as well.
Why does the parabola not cross the x-axis when D < 0?
When D < 0, the roots are complex, so there are no real x-intercepts. The parabola lies entirely above or below the x-axis; the graph shows the curve and a note that roots are complex.
What is the vertex of a parabola?
For f(x) = ax² + bx + c, the vertex is (h, k) with
and
The axis of symmetry is x = h. The calculator shows the vertex and draws the axis on the parabola
How do I solve a quadratic equation with the calculator?
Enter the coefficients a, b, and c (a ≠ 0). The calculator computes the discriminant and roots, and updates the parabola and Logic Trace. Use Copy results to export the formula, roots, and vertex.