📐
Quadratic Formula Calculator
Solve ax² + bx + c = 0 using the quadratic formula
Share Quadratic Formula
📚 How to Use & Examples▼
⚡Quick Examples - Try These Calculations
🔍How It Works
Quadratic Formula
For expression ax² + bx + c = 0, use the quadratic formula:
x=
-b±b²-4ac
2a
Discriminant Rules
The discriminant determines solution types:
Discriminant=b²-4ac
- • If positive: two real solutions
- • If zero: one solution (repeated)
- • If negative: complex solutions
Vertex Formula
The vertex of parabola y = ax² + bx + c is at:
x=
-b
2a
Substitute this x back into the equation to find the y-coordinate.
🌍Real-World Applications
⚡ Physics
Projectile motion, wave analysis, energy calculations
🏗️ Engineering
Structural design, signal processing, optimization
💰 Economics
Profit maximization, cost analysis, revenue optimization
💻 Computer Science
Algorithm complexity, graphics, machine learning
💡Calculator Tips & Best Practices
💡Check Your Discriminant
Always calculate the discriminant first to know what type of solutions to expect
⚠️Coefficient 'a' Cannot Be Zero
If a = 0, you have a linear expression, not a quadratic expression
⭐Verify Solutions
Always substitute your solutions back into the original expression to verify they're correct
💡Factoring Alternative
If the quadratic factors easily, factoring might be faster than using the quadratic formula
❓Frequently Asked Questions
What if coefficient 'a' is zero?
If a = 0, the expression becomes linear (bx + c = 0), not quadratic. The coefficient 'a' must be non-zero for a quadratic expression.
What do complex solutions mean?
Complex solutions occur when the discriminant is negative. These solutions involve the imaginary unit 'i' and represent points where the parabola doesn't cross the x-axis.
How do I verify my solutions?
Substitute each solution back into the original equation. If both sides equal zero, your solution is correct.
What's the significance of the vertex?
The vertex is the turning point of the parabola. It's the minimum point if a > 0 (opens upward) or maximum point if a < 0 (opens downward).
📚 References & Further Reading
Comprehensive coverage of quadratic formulas and their applications
External Link
Interactive lessons on solving quadratic expressions
External Link
Note: These references provide additional mathematical context and verification of the formulas used in this calculator.