Quadratic Equation Solver
Solve quadratic equations (ax² + bx + c = 0) using the quadratic formula. Shows discriminant, roots (real or complex), and vertex form.
What is Quadratic Equation Solver?
This quadratic equation solver finds both roots using the quadratic formula x = (−b ± √(b²−4ac)) / 2a. Determines if roots are real and distinct, real and equal, or complex conjugates based on the discriminant. Also provides vertex form and axis of symmetry.
How to Use This Calculator
- Enter your a (coefficient of x²), b (coefficient of x), c (constant) in the input fields provided
- Results are computed instantly as you enter or modify values — no need to click a button
- Review the computed output showing your quadratic equation solver results with a detailed breakdown
- Cross-check the output with your manual working to reinforce the underlying concept
How Quadratic Equation Solver is Calculated
This calculator uses the formula: x = (−b ± √(b² − 4ac)) / 2a. Where a = coefficient of x² ; b = coefficient of x ; c = constant ; Δ = b²−4ac (discriminant). Quadratic formula solves ax²+bx+c=0. Discriminant determines: 2 real, 1 repeated, or complex roots. All calculations run entirely in your browser — no data is transmitted to any server.
Frequently Asked Questions
What does the discriminant tell us?
D = b² − 4ac. If D > 0: two distinct real roots. D = 0: one repeated real root (vertex touches x-axis). D < 0: two complex conjugate roots (parabola does not cross x-axis).
How do I find the vertex of a parabola?
Vertex x-coordinate: x = −b/(2a). Substitute back to find y. For y = 2x² − 8x + 3: vertex x = 8/4 = 2, y = 2(4) − 16 + 3 = −5. Vertex: (2, −5). This is the minimum point when a > 0.
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Why You Need This Calculator
Quadratic equations (ax² + bx + c = 0) appear everywhere — from physics projectile motion to business break-even analysis. The discriminant (b² – 4ac) determines whether solutions are real, equal, or complex. This calculator provides step-by-step solutions using the quadratic formula, completing the square, and graphical representation.
Calculator Features
- Solve any quadratic equation instantly
- Step-by-step solution method
- Discriminant analysis (real/complex roots)
- Graphical representation with vertex
- Sum and product of roots
- Nature of roots classification
The Math Behind It
x = [-b ± √(b² – 4ac)] / 2a. Discriminant D = b² – 4ac. Vertex: h = -b/2a, k = f(h).
Calculation Example
2x² + 5x – 3 = 0. D = 25 + 24 = 49. x = (-5 ± 7) / 4. x₁ = 2/4 = 0.5, x₂ = -12/4 = -3. Verify: sum = -5/2 ✓, product = -3/2 ✓.
Quick Reference
Quadratic equation examples by discriminant type
| Equation | Discriminant | Nature | Roots |
|---|---|---|---|
| x² – 5x + 6 = 0 | 1 | Two real | x = 2, 3 |
| x² – 4x + 4 = 0 | 0 | Repeated | x = 2 |
| x² + x + 1 = 0 | -3 | Complex | x = (-1 ± i√3)/2 |
Pro Tips & Expert Insights
- 💡 If discriminant > 0: two distinct real roots. = 0: one repeated root. < 0: complex conjugate roots.
- 💡 Sum of roots = -b/a. Product of roots = c/a. Use this to verify your answers.
- 💡 For physics problems, negative roots often represent "before the event started."
- 💡 Factor when possible — it's faster than the formula for simple equations.
- 💡 The vertex form a(x-h)² + k reveals the maximum/minimum directly.
Who Benefits From This?
Students studying algebra, JEE/NEET aspirants, physics students solving kinematics, and engineers.
📚 Complete Guide Available
Want to learn more? Read our comprehensive guide with detailed explanations, real-world examples, expert analysis, and actionable tips.
Read: QuadraticNote: This calculator provides results based on standard mathematical formulas. Always verify important calculations independently.
Maintained by: Sagar Sahni, Calc Labz | Review: formula checks, worked examples, and periodic updates
Need a correction? Contact us with the calculator name, your inputs, and the issue you found.
Last updated: April 2026