Quadratic Formula Calculator
Solve Quadratic Equation
Enter coefficients a, b, and c for ax² + bx + c = 0
Results & Actions
Calculate and view quadratic solutions
Step-by-Step Solution
- Equation: 1x² + 0x + 0 = 0
- Discriminant: Δ = b² - 4ac = 0² - 4×1×0 = 0
- Δ = 0: One real repeated root
- Root: x = -b / 2a = 0 / 2 = 0
Parabola Visualization
Vertex: (0, 0)
Axis: x = 0
Direction: Upward
How It Works
Enter coefficients a, b, and c to solve ax² + bx + c = 0. The calculator computes discriminant, determines root type, and provides exact solutions with step-by-step explanations.
Root Types
Based on discriminant value: Δ > 0 gives two real roots, Δ = 0 gives one real repeated root, Δ < 0 gives complex conjugate roots with imaginary components.
Real Applications
Used in physics for projectile motion, engineering for optimization problems, architecture for parabolic designs, economics for profit maximization, and computer graphics.
Quadratic Formula & Equations
Standard Form
Quadratic Formula
Discriminant
Step-by-Step Examples
Projectile Motion
Δ = 20² - 4(-4.9)(1) = 419.6
t = [-20 ± √419.6] / -9.8
Area Optimization
Δ = (-12)² - 4(1)(35) = 4
x = [12 ± √4] / 2
Complex Roots
Δ = 2² - 4(1)(5) = -16
x = [-2 ± √(-16)] / 2
Understanding Quadratic Equations
Quadratic equations are polynomial equations of degree 2 that describe parabolic relationships in mathematics, physics, engineering, and many other fields. The graph of a quadratic equation is always a parabola, which can open upward or downward depending on the coefficient 'a'.
What is a Quadratic Equation?
A quadratic equation is any equation that can be rearranged in standard form as ax² + bx + c = 0, where x represents an unknown variable, and a, b, and c represent known numbers with a ≠ 0. The solutions to these equations are called roots or zeros of the equation.
The Discriminant: Key to Understanding Roots
The discriminant (Δ = b² - 4ac) determines the nature of the roots without actually calculating them:
- Δ > 0: Two distinct real roots - the parabola crosses the x-axis at two different points
- Δ = 0: One real repeated root - the parabola touches the x-axis at exactly one point (the vertex)
- Δ < 0: Two complex conjugate roots - the parabola doesn't intersect the x-axis at any real points
Practical Applications
Quadratic equations model numerous real-world phenomena:
Physics: Projectile Motion
The height h of a projectile at time t is given by: h(t) = -½gt² + v₀t + h₀
Solving h(t) = 0 gives the times when the projectile is at ground level. For example, a ball thrown upward from 1 meter with initial velocity 20 m/s under gravity (g = 9.8 m/s²) follows: -4.9t² + 20t + 1 = 0
The solutions t = 0.05s and t = 4.13s represent when the ball passes through height 1 meter on its way up and down.
Geometry: Area Optimization
If a rectangle has area 35 square units and perimeter 24 units, its dimensions satisfy: x² - 12x + 35 = 0
The solutions x = 7 and x = 5 give the rectangle's dimensions. This demonstrates how quadratic equations solve optimization problems where you maximize or minimize quantities under constraints.
How to Interpret the Results
Real Roots: Represent actual x-intercepts where the parabola crosses the x-axis. These are the solutions that make the equation true.
Complex Roots: Indicate the parabola doesn't intersect the x-axis. In practical terms, this means no real number satisfies the equation, which can occur in systems that don't reach certain states.
Vertex: The turning point of the parabola (minimum if a > 0, maximum if a < 0). Calculated at x = -b/(2a), this point represents optimal values in optimization problems.