Quadratic Equation Solver
Solve quadratic equations using the quadratic formula
Enter the coefficients of the quadratic equation in the form: ax² + bx + c = 0
The Complete Guide to Solving Quadratic Equations
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving these equations helps in:
- Finding the x-intercepts of a parabola
- Modeling physical phenomena (e.g., projectile motion)
- Analyzing optimization problems
Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
Key Formulas for Solving Quadratic Equations
Quadratic Formula
x = (-b ± √(b² - 4ac)) / (2a)
This formula computes the roots of the equation by substituting the values of a, b, and c.
Discriminant
Discriminant: Δ = b² - 4ac
The discriminant determines the nature of the roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root
- Δ < 0: Two complex roots
How to Use the Quadratic Equation Solver: Step-by-Step
Step 1: Input Coefficients
- Enter the values for a, b, and c in the form ax² + bx + c = 0.
- Ensure a ≠ 0; otherwise, the equation is not quadratic.
Step 2: Interpret the Results
The solver calculates:
- Roots (x₁ and x₂), which may be real or complex
- The discriminant (Δ) for understanding root nature
Step 3: Verify and Apply
Use the calculated roots to solve real-world problems, such as finding maximum heights, optimization, or intersections in graphs.
Practical Tips for Solving Quadratic Equations
Check for Simplification
- Simplify the equation before solving (e.g., remove common factors).
- Verify coefficient values to ensure accuracy.
Interpret Roots Correctly
- Real roots indicate x-intercepts of the parabola.
- Complex roots show no real x-intercepts.
Quadratic Equation FAQs
What does the discriminant tell us about the roots?
The discriminant (Δ = b² - 4ac) determines the type of roots: two real roots if Δ > 0, one repeated root if Δ = 0, or two complex roots if Δ < 0.
Can the quadratic formula solve all quadratic equations?
Yes, the quadratic formula works for all quadratic equations as long as the coefficients are real numbers and a ≠ 0.