Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula.
Enter Coefficients for ax² + bx + c = 0
x₂ = 2
Step-by-Step Solution:
1. Discriminant: b² - 4ac = 25 - 24 = 1
2. x = [-b ± √(b² - 4ac)] / 2a
3. x = [5 ± √1] / 2
4. x = [5 ± 1] / 2
5. x₁ = (5 + 1) / 2 = 3
6. x₂ = (5 - 1) / 2 = 2
The Quadratic Formula
For equations in the form ax² + bx + c = 0:
x = [-b ± √(b² - 4ac)] / 2a
How to Use the Quadratic Formula
- Identify a, b, and c from your equation ax² + bx + c = 0
- Calculate the discriminant: b² - 4ac
- Apply the formula: x = [-b ± √(b² - 4ac)] / 2a
- Simplify to find both solutions
Understanding the Discriminant
The discriminant (b² - 4ac) tells you about the solutions:
- Positive: Two different real solutions
- Zero: One real solution (repeated root)
- Negative: Two complex solutions
Real-World Applications
Quadratic equations model: - Projectile motion (physics) - Area and optimization problems - Profit and revenue calculations - Engineering and design - Computer graphics
Common Questions
What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a. It solves any quadratic equation in the form ax² + bx + c = 0.
What does the discriminant tell you?
The discriminant (b² - 4ac) determines the nature of solutions: positive = two real solutions, zero = one solution, negative = two complex solutions.