Quadratic Formula Calculator
Solve Quadratic Equations
Enter coefficients a, b, and c to solve quadratic equations using the quadratic formula.
Solution Results
x = 1, 2
Discriminant Analysis:
The quadratic formula provides solutions to equations of the form ax² + bx + c = 0.
What is the Quadratic Formula?
The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0. It provides the roots (solutions) of the equation by considering all possible cases of the discriminant.
Quadratic Formula Components
Discriminant
Determines the nature of roots
Real Roots
When D ≥ 0
Complex Roots
When D < 0
Vertex
Turning point of parabola
Nature of Roots Based on Discriminant
1. Positive Discriminant (D > 0)
Two distinct real roots:
x = [-b + √D]/2a, [-b - √D]/2a
2. Zero Discriminant (D = 0)
One real repeated root:
x = -b/2a
3. Negative Discriminant (D < 0)
Two complex conjugate roots:
x = [-b ± i√|D|]/2a
Real-World Applications
Physics & Engineering
- Projectile motion calculations
- Spring and oscillation systems
- Electrical circuit analysis
- Optics and lens equations
Economics & Business
- Profit maximization problems
- Cost and revenue analysis
- Market equilibrium models
- Investment return calculations
Example Calculations
Example 1: Two Real Roots
Equation: x² - 5x + 6 = 0
a = 1, b = -5, c = 6
D = (-5)² - 4(1)(6) = 25 - 24 = 1
Roots: x = [5 ± 1]/2 = 3, 2
Example 2: Complex Roots
Equation: x² + 4x + 13 = 0
a = 1, b = 4, c = 13
D = 4² - 4(1)(13) = 16 - 52 = -36
Roots: x = [-4 ± 6i]/2 = -2 ± 3i
Discriminant Analysis Table
| Discriminant | Nature of Roots | Graphical Meaning |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola intersects x-axis at two points |
| D = 0 | One real repeated root | Parabola touches x-axis at one point |
| D < 0 | Two complex roots | Parabola doesn't intersect x-axis |
Frequently Asked Questions (FAQs)
Q: What if coefficient a is zero?
A: If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula doesn't apply.
Q: Can quadratic equations have more than two solutions?
A: No, by the Fundamental Theorem of Algebra, a quadratic equation has exactly two solutions.
Q: How do I know if roots are rational?
A: Roots are rational when the discriminant is a perfect square and coefficients are rational numbers.
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