Quadratic Equation Solver 2026 – Solve ax²+bx+c=0
Solve any quadratic equation ax² + bx + c = 0 instantly. Get exact roots with the quadratic formula, factored form, discriminant analysis, vertex coordinates, and clear step-by-step working — all computed in your browser.
Quadratic Equation Calculator
Enter the coefficients a, b, and c. Calculations run in your browser.
What Is a Quadratic Equation Solver?
A quadratic equation solver turns the quadratic formula into a quick browser-based calculation. Instead of working through the discriminant and square roots by hand, you enter the coefficients a, b, and c and immediately see the roots, the factored form, the vertex, and every substitution step.
How to Use the Quadratic Equation Solver
The tool has three modes so you can start with the question you need answered.
- Enter a, b, c for the equation ax² + bx + c = 0 (a must not be zero).
- Choose a mode that matches your question.
- Select Calculate to generate the result.
- Check the result cards for roots, discriminant, steps, factored form, vertex, and axis.
The solver runs in your browser and shows exact fractions where roots are rational. It is intended for homework checks, practice, and verifying manual work.
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions (roots) are the x-values where the parabola y = ax² + bx + c crosses the x-axis.
Worked Examples
| Equation | Discriminant | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | D = 25 − 24 = 1 | x = 2, x = 3 |
| 2x² − 7x + 3 = 0 | D = 49 − 24 = 25 | x = 3, x = 1/2 |
| x² + 2x + 1 = 0 | D = 4 − 4 = 0 | x = −1 (double root) |
| x² + x + 1 = 0 | D = 1 − 4 = −3 | x = (−1 ± i√3)/2 |
The Quadratic Formula
The quadratic formula gives the roots of any quadratic equation directly from its coefficients:
Substitute a, b, and c, simplify the discriminant, then evaluate the two branches (−b + √D) / 2a and (−b − √D) / 2a.
Discriminant Table
| Discriminant D | Root Type | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x² − 5x + 6 = 0 → x = 2, 3 |
| D = 0 | One repeated real root | x² + 2x + 1 = 0 → x = −1 |
| D < 0 | Two complex conjugate roots | x² + 1 = 0 → x = ±i |
Step-by-Step: x² − 5x + 6 = 0
- Identify coefficients: a = 1, b = −5, c = 6.
- Compute the discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1.
- Take the square root: √D = 1.
- Apply the formula: x = (5 ± 1) / 2.
- Roots: x = 3 and x = 2.
Solving by Factoring
Factoring rewrites the trinomial as a product of two binomials, then uses the zero-product property: if AB = 0, then A = 0 or B = 0.
Step-by-Step: x² − 5x + 6 = 0
- Find two numbers that multiply to c = 6 and add to b = −5: −2 and −3.
- Write the factored form: (x − 2)(x − 3) = 0.
- Set each factor to zero: x − 2 = 0 or x − 3 = 0.
- Roots: x = 2 and x = 3.
Common Mistakes When Solving Quadratics
1. Forgetting a ≠ 0
If a = 0, the equation is linear, not quadratic. The quadratic formula requires division by 2a, so it does not apply.
2. Sign errors with −b
When b is negative, −b becomes positive. For x² − 5x + 6 = 0, −b = 5, not −5.
3. Dropping the ±
The ± gives two roots. Forgetting it loses half the solution.
4. Miscomputing the discriminant
D = b² − 4ac — watch the signs. (−5)² = 25, and −4(1)(6) = −24, so D = 1.
5. Taking √D incorrectly for D < 0
A negative discriminant means complex roots, not "no solution." x² + 1 = 0 gives x = ±i.
6. Dividing by a before checking a = 0
Always confirm a ≠ 0 first; otherwise you may divide by zero.
7. Confusing factored form with roots
(x − 2)(x − 3) = 0 gives roots x = 2 and x = 3 — the signs flip.
8. Rounding too early
Keep exact values (fractions, radicals) through the steps; round only the final decimal.
Benefits & Limitations
Limitations
- Only quadratic equations (degree 2) are supported — cubic and higher-degree polynomials need a different solver.
- Factoring mode finds integer factor pairs; non-integer roots are better handled by the quadratic formula mode.
- Very large coefficients may lose floating-point precision in the decimal display; exact forms are preferred.
Quick Definitions
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and its solutions are the x-intercepts.
How do you solve a quadratic equation?
Use the quadratic formula x = (−b ± √(b²−4ac)) / 2a, factor the trinomial into (x − r₁)(x − r₂) = 0, or complete the square. The tool above shows every step.
What is the discriminant?
The discriminant D = b² − 4ac tells how many real roots exist: D > 0 gives two distinct real roots, D = 0 gives one repeated root, and D < 0 gives two complex roots.
What happens when a = 0?
If a = 0, the equation becomes bx + c = 0, which is linear — not quadratic. The quadratic formula does not apply because it divides by 2a.
How do you find the vertex of a parabola?
The vertex is at h = −b/2a and k = f(h) = a·h² + b·h + c. The axis of symmetry is the vertical line x = h.
Can a quadratic have no real roots?
Yes. When the discriminant is negative, the roots are a complex conjugate pair. For example, x² + 1 = 0 gives x = i and x = −i, so there are no real roots.
Methodology
This page was produced by combining a review of standard algebra references (Khan Academy, Paul's Online Math Notes, Math is Fun, Wolfram MathWorld, Purplemath) with direct verification of every worked example using the quadratic formula, factoring, and discriminant analysis. Each numerical claim on this page — including all four worked examples and the discriminant table — was recomputed by hand and cross-checked with the browser-based solver above before publication.
References
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