Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula with step-by-step solutions and detailed analysis
Solve Quadratic Equation
Standard Form: Ax² + Bx + C = 0
Coefficient of x² (cannot be 0 for quadratic)
Coefficient of x (can be 0)
Constant term (can be 0)
Solution Results
Vertex of Parabola
Coordinates: (0.0000, 0.0000)
Axis of Symmetry: x = 0.0000
Vertex Type: Minimum point
Function Forms
Standard Form: f(x) = 1x² + 0x + 0
Vertex Form: f(x) = 1(x - (0.0000))² + (0.0000)
Step-by-Step Solution
Given quadratic equation: x² = 0
Coefficients: A = 1, B = 0, C = 0
Calculate discriminant: Δ = B² - 4AC
Δ = (0)² - 4(1)(0)
Δ = 0 - 0 = 0
Since Δ = 0, the equation has one repeated real root
Apply quadratic formula: x = -B / (2A)
x = -(0) / (2 × 1)
x = 0 / 2 = 0.000000
Example Problems
Two Real Roots
Equation: x² - 5x + 6 = 0
Coefficients: A = 1, B = -5, C = 6
Discriminant: Δ = 25 - 24 = 1
Roots: x₁ = 3, x₂ = 2
One Repeated Root
Equation: x² - 4x + 4 = 0
Coefficients: A = 1, B = -4, C = 4
Discriminant: Δ = 16 - 16 = 0
Root: x = 2 (double root)
Complex Roots
Equation: x² + 2x + 5 = 0
Coefficients: A = 1, B = 2, C = 5
Discriminant: Δ = 4 - 20 = -16
Roots: x₁ = -1 + 2i, x₂ = -1 - 2i
Quadratic Formula
A: Coefficient of x² (≠ 0)
B: Coefficient of x
C: Constant term
Δ: Discriminant
Discriminant Analysis
Δ > 0
Two distinct real roots
Parabola crosses x-axis twice
Δ = 0
One repeated real root
Parabola touches x-axis once
Δ < 0
No real roots (complex)
Parabola doesn't touch x-axis
Quick Tips
A must be ≠ 0 for a quadratic equation
When A > 0, parabola opens upward
When A < 0, parabola opens downward
Sum of roots = -B/A
Product of roots = C/A
Understanding the Quadratic Formula
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in the standard form Ax² + Bx + C = 0, where A, B, and C are constants and A ≠ 0. The graph of a quadratic equation is a parabola.
The Quadratic Formula
The quadratic formula x = (-B ± √Δ)/(2A) provides the exact solutions to any quadratic equation. It's derived by completing the square on the general form of a quadratic equation.
Applications
- •Physics: Motion equations, projectile paths
- •Engineering: Optimization problems
- •Economics: Cost and revenue functions
- •Architecture: Parabolic structures
How to Use This Calculator
The Golden Ratio Connection
The famous golden ratio φ = (1 + √5)/2 ≈ 1.618 comes from solving the quadratic equation x² - x - 1 = 0. This ratio appears in art, architecture, and nature.
Fun Fact: The quadratic formula was known to ancient Babylonians around 2000 BCE, though they didn't express it in our modern algebraic notation!