🎓 Mathematics • Algebra
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Quadratic Equation Solver
Solves quadratic equations ax² + bx + c = 0, calculating discriminant Δ, real and complex roots, and parabola vertex.
Algebra & Matrix Engine
Algebra & Matrix Studio
Solve quadratic equations with complex roots, 2x2 systems via Cramer's rule, binomial FOIL expansions, and 2x2/3x3 matrix linear algebra with instant proofs.
Quadratic Coefficients
Standard form: ax² + bx + c = 0 (where a ≠ 0).
Sample Presets:
Parabola Geometry
Vertex (h, k)(2.5, -0.25)
Axis of Symmetryx = 2.5
OrientationOpens Upward (Minimum)
Discriminant (Δ)1
Two Distinct Real Roots
1x² - 5x + 6 = 0
Root x1
3
Root x2
2
Factored Form: (x - 3)(x - 2) = 0
Step-by-Step Derivation (Quadratic Formula)
1Equation: 1x² - 5x + 6 = 0
2Discriminant Δ = b² - 4ac = (-5)² - 4(1)(6) = 1
3Δ > 0: Two distinct real roots.
4x₁ = (-(-5) + √1) / (2 × 1) = 3
5x₂ = (-(-5) - √1) / (2 × 1) = 2
6Parabola Vertex (h, k) = (-b/2a, c - b²/4a) = (2.5, -0.25)
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation. Its graph forms a U-shaped parabola.
The quadratic formula provides roots by evaluating the discriminant Δ = b² - 4ac.
Formula & Step-by-Step Calculation
x = (-b ± √(b² - 4ac)) / (2a)
Universal formula for real or complex roots of any quadratic equation.
Worked Step-by-Step Examples
Example 1
Solve x² - 9 = 0
Solution: x = ±3
• x² = 9 ⟹ x = ±3
Common Real-World & Academic Use Cases
- ✓ Projectile height and time of flight
- ✓ Maximizing profit parabolas
- ✓ Engineering mechanics
How to Use the Quadratic Equation Solver
1
Input a, b, c
Enter quadratic coefficients.
2
Read Roots
View real or complex roots and vertex.
Frequently Asked Questions
Q: What does Δ < 0 mean?
Negative discriminant indicates two complex conjugate roots containing imaginary unit i.