MathematicsAlgebra
Quadratic Equation Solver & Formula Calculator
Solve any quadratic equation ax² + bx + c = 0. Computes real or complex imaginary roots, discriminant value, and parabola vertex coordinates.
Coefficients: ax² + bx + c = 0
x² - 5x + 6 = 0
Quadratic Roots & Vertex
Two distinct real roots: x₁ = 3, x₂ = 2
Discriminant (Δ = b² - 4ac): 1
Parabola Vertex (h, k):(2.5, -0.25)
Axis of Symmetry:x = 2.5
Mathematical Formula: The Quadratic Formula
x = [-b ± √(b² - 4ac)] / (2a)
Finds the x-intercepts (roots) of any second-degree polynomial equation where a ≠ 0.
Variable Definitions:
- a= Quadratic coefficient (cannot be 0)
- b= Linear coefficient
- c= Constant term
- Δ= Discriminant: b² - 4ac
How to Use This Calculator
- 1Enter the coefficients a, b, and c from your equation.
- 2The solver calculates the discriminant Δ = b² − 4ac.
- 3Outputs distinct real roots (Δ > 0), repeated root (Δ = 0), or complex conjugate roots (Δ < 0).
- 4Provides the parabola vertex point (h, k).
Practical Example: Solving x² - 5x + 6 = 0
Scenario: a = 1, b = -5, c = 6
a:1
b:-5
c:6
Result: Roots: x₁ = 3, x₂ = 2 | Discriminant: 1 | Vertex: (2.5, -0.25)
Discriminant is positive (25 - 24 = 1), yielding two distinct real solutions.
Frequently Asked Questions
If b² - 4ac > 0, there are 2 real roots. If = 0, there is 1 real root. If < 0, there are 2 complex (imaginary) roots.
Pro Calculation Tips
- If the discriminant is a perfect square (1, 4, 9, 16, 25...), the quadratic can be factored into rational brackets.
Model Assumptions
- Coefficient "a" cannot be 0, as that reduces to a linear equation.