Solve any quadratic equation with a full step-by-step breakdown — discriminant, nature of roots, sum/product of roots, vertex, and a graph — or build an equation from two known roots.
Key Takeaways
- The discriminant alone tells you what kind of roots to expect before you even finish solving — positive means two real roots, zero means one repeated real root, negative means a complex conjugate pair.
- Sum and product of roots come straight from the coefficients, without solving the equation at all: sum = −b/a, product = c/a — useful for checking an answer or building an equation backward from known roots.
- A quadratic's graph is always a parabola, and its real roots (if any) are exactly where that parabola crosses the x-axis — a negative discriminant means the parabola never touches the x-axis at all.
What This Quadratic Equation Solver Does
This tool solves any equation of the form ax² + bx + c = 0 using the quadratic formula, and shows the full picture, not just the final answer: the discriminant and what it tells you about the roots, the sum and product of the roots, the vertex and axis of symmetry, and a graph of the parabola with its roots and vertex marked. It handles real roots (distinct or repeated) and complex roots equally, showing complex answers in standard a + bi form. A second mode works backward — given two roots, or just their sum and product, it builds the simplest matching quadratic equation, which is exactly the reverse skill tested alongside solving in most Class 10 curricula.
Everything runs entirely in your browser. No equation or coefficient you enter is sent to a server, logged, or stored anywhere — close the tab and it's gone. There's no signup, no email wall, and no ads inside the calculator itself. Use it to check your own working, explore how changing a coefficient shifts the graph, or quickly solve an equation you don't want to work through by hand.
How to Use This Quadratic Equation Solver
1. Solve an Equation
- Enter the coefficients a, b, and c from ax² + bx + c = 0.
- Read the roots, discriminant, and nature of the roots.
- Check the step-by-step working and the graph.
2. Build an Equation from Roots
- Switch to Build Equation from Roots.
- Enter the two roots, or their sum and product.
- Read the resulting equation.
The Quadratic Formula, Explained
For any equation ax² + bx + c = 0 (with a ≠ 0), the roots are:
x = [−b ± √(b² − 4ac)] / 2a
The expression under the square root, D = b² − 4ac, is the discriminant. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, the roots are a complex conjugate pair.
Worked example: x² − 5x + 6 = 0, so a = 1, b = −5, c = 6.
- Discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1
- Since D > 0, two real distinct roots
- x = [5 ± √1] / 2 = (5 ± 1) / 2
- Roots: x = 3 and x = 2
Check using sum and product of roots: Sum = −b/a = 5 (and 3+2=5, correct); Product = c/a = 6 (and 3×2=6, correct).
Discriminant & Nature of Roots Reference Table
What the sign of the discriminant tells you before you finish solving.
| Discriminant (D = b² − 4ac) | Nature of Roots | Graph Behaviour |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| D = 0 | One repeated real root | Parabola touches the x-axis at exactly one point (the vertex) |
| D < 0 | Two complex conjugate roots | Parabola never touches the x-axis |
Quick Definitions
The discriminant of a quadratic equation is the value b² − 4ac, which determines how many real solutions the equation has without needing to fully solve it. The vertex of a parabola y = ax² + bx + c is its highest or lowest point, located at x = −b/2a, and is the point around which the parabola is symmetric (the axis of symmetry).
Frequently Asked Questions
If a = 0, the equation is no longer quadratic — it becomes a linear equation, bx + c = 0, which has at most one solution instead of two. This calculator requires a non-zero value for a.
Complex roots show up whenever a quadratic's graph doesn't cross the x-axis, which happens often in higher-level algebra, electrical engineering (analysing AC circuits), and control systems. In a typical school-level word problem about real-world quantities, a negative discriminant usually signals that the scenario as posed has no real solution.
For ax² + bx + c = 0, the sum of the roots always equals −b/a and the product always equals c/a, regardless of whether the roots are real or complex. This relationship is useful both for quickly checking a solved answer and for building an equation when only the roots (or their sum and product) are known.
No, this tool is specifically for quadratic (degree 2) equations. Cubic and higher-degree polynomials need different solving methods and aren't covered by the quadratic formula.
Infinitely many quadratics share the same two roots — multiplying every coefficient by the same non-zero number doesn't change where the roots are. Using a = 1 gives the simplest such equation, in the form x² − (sum of roots)x + (product of roots) = 0.
No. Every calculation runs locally in your browser using JavaScript; nothing you enter is transmitted to any server, logged, or saved once you close or refresh the page.