Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Get roots, discriminant, vertex, and step-by-step solutions.
Enter Quadratic Equation
ax² + bx + c = 0
Enter the coefficients a, b, and c
Results
Enter coefficients and click "Solve Equation" to see the results
Understanding the Quadratic Formula
What is the Quadratic Formula?
The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax² + bx + c = 0, where a ≠ 0.
x = (-b ± √(b² - 4ac)) / (2a)
The Discriminant
The discriminant (b² - 4ac) determines the nature of the roots:
- Positive: Two distinct real roots
- Zero: One repeated real root
- Negative: Two complex conjugate roots
Step-by-Step Solution Process
- 1. Identify coefficients: Find the values of a, b, and c from the equation ax² + bx + c = 0
- 2. Calculate discriminant: Compute b² - 4ac to determine the nature of roots
- 3. Apply the formula: Use x = (-b ± √(b² - 4ac)) / (2a)
- 4. Simplify: Calculate both roots using the plus and minus operations
- 5. Verify: Substitute the roots back into the original equation to check
Real-World Applications
- • Projectile motion
- • Area optimization
- • Profit maximization
- • Engineering design
- • Physics calculations
Types of Solutions
- • Two distinct real roots
- • One repeated root
- • Complex conjugate pairs
- • Rational vs irrational
- • Integer solutions
Related Concepts
- • Vertex form
- • Factoring quadratics
- • Completing the square
- • Parabola properties
- • Axis of symmetry
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