Quadratic Equation Calculator
Solve any quadratic equation ax² + bx + c = 0 in seconds. Enter the three coefficients, pick your method (quadratic formula, factoring, completing the square, or all three), and get the roots with full step-by-step working, the discriminant, the vertex, the axis of symmetry, and a graph of the parabola with the roots marked.
Roots, steps, vertex, and graph for ax² + bx + c = 0. Real and complex roots supported.
Leave b or c blank to treat them as zero. The factoring method only applies when the roots are integers; the tool says so honestly when they are not.
Worked example: 2x² + 5x − 3 = 0
Enter a = 2, b = 5, c = −3, choose Show all methods, and click Solve equation. The discriminant is D = 25 − 4(2)(−3) = 49, so there are two distinct real roots: x₁ = 0.50 and x₂ = −3.00. The vertex sits at (−1.25, −6.125), the axis of symmetry is x = −1.25, the y-intercept is −3, and the sum and product of the roots are −2.5 and −1.5, matching −b/a and c/a. The graph shows the parabola crossing the x-axis exactly at 0.5 and −3. These are the tool's real outputs, verified against its code.
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / (2a). The discriminant D = b² − 4ac decides the story before you compute anything: D > 0 gives two distinct real roots, D = 0 gives one repeated root, and D < 0 gives two complex conjugate roots. The vertex of the parabola y = ax² + bx + c is at x = −b/(2a), which is also the axis of symmetry.
Choosing a solution method
The quadratic formula always works and is the default. Factoring is fastest when the roots are integers, but it honestly reports "cannot factor with integer coefficients" for equations like x² − 2 = 0 instead of faking it. Completing the square rewrites the equation as (x + b/2)² = (b/2)² − c, the form that reveals the vertex directly. Show all methods prints every applicable working side by side, which is the fastest way to compare them for homework.
Reading the graph
The SVG graph plots y = ax² + bx + c with the vertex marked in red and each real root marked in green where the curve crosses the x-axis. When a > 0 the parabola opens upward and the vertex is the minimum; when a < 0 it opens downward and the vertex is the maximum. Complex roots mean the curve never touches the x-axis, which the graph makes obvious at a glance.
What competing calculators miss
Most quadratic tools print two roots and stop: no vertex, no axis, no method choice, no steps. The strongest competitor offers method and precision selects but only claims graph support without clearly rendering one. This calculator keeps all of that (formula, factoring with an honest fallback, completing the square, exact mode) and adds what none of them show together: a real rendered parabola with the vertex and roots marked, plus the y-intercept and the sum and product of roots as built-in answer checks.
Frequently asked questions
How do I solve a quadratic equation?
Identify a, b, and c in ax² + bx + c = 0, compute the discriminant b² − 4ac, then apply x = (−b ± √D) / (2a). The calculator does all of this and shows every step.
What does the discriminant tell me?
Whether the roots are real or complex before you find them: positive means two distinct real roots, zero means one repeated root, negative means two complex conjugate roots.
Can a quadratic have complex roots?
Yes. When the discriminant is negative, the roots are a conjugate pair like −1 + 2i and −1 − 2i. The graph then never crosses the x-axis.
What if my equation cannot be factored?
Choose the quadratic formula or completing the square instead. The factoring method here only claims success for integer roots and tells you when it does not apply.
What is the vertex of a parabola?
The turning point, at x = −b/(2a). It is the minimum when a > 0 and the maximum when a < 0, and the vertical line through it is the axis of symmetry.
How do I check my roots?
Use the root relationships: the sum of the roots equals −b/a and the product equals c/a. The calculator prints both so you can verify instantly.