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.