Quadratic Formula Calculator - Solve ax²+bx+c=0 Equations Online
Quadratic Formula: x = (-b ± √(b²-4ac)) / 2a
Enter coefficients for ax² + bx + c = 0
Solution:
Equation: 1x² + 0x + 0 = 0
Discriminant: 0
Type: One real root (repeated)
Root 1: x₁ = 0
How the Quadratic Formula Works
Input Coefficients
Enter values for a, b, c
Calculate Discriminant
Δ = b² - 4ac
Apply Formula
x = (-b ± √Δ) / 2a
Step-by-Step Solution
Given: 1x² + 0x + 0 = 0
Discriminant = b² - 4ac = 0² - 4(1)(0) = 0
Since discriminant = 0, there is one repeated real root
x = -b / 2a = 0 / 2 = 0.000000
Quadratic Equation Examples
| Equation | a | b | c | Discriminant | Roots |
|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | 1 | x = 2, 3 |
| x² - 4x + 4 = 0 | 1 | -4 | 4 | 0 | x = 2 (repeated) |
| x² + x + 1 = 0 | 1 | 1 | 1 | -3 | x = -0.5 ± 0.866i |
| 2x² - 7x + 3 = 0 | 2 | -7 | 3 | 25 | x = 3, 0.5 |
| x² - 6x + 9 = 0 | 1 | -6 | 9 | 0 | x = 3 (repeated) |
| 3x² + 2x - 1 = 0 | 3 | 2 | -1 | 16 | x = 0.333, -1 |
| x² + 2x + 5 = 0 | 1 | 2 | 5 | -16 | x = -1 ± 2i |
| 4x² - 4x + 1 = 0 | 4 | -4 | 1 | 0 | x = 0.5 (repeated) |
| x² - 3x - 4 = 0 | 1 | -3 | -4 | 25 | x = 4, -1 |
| 2x² + 3x - 2 = 0 | 2 | 3 | -2 | 25 | x = 0.5, -2 |
| x² + 4x + 13 = 0 | 1 | 4 | 13 | -36 | x = -2 ± 3i |
| 5x² - 10x + 5 = 0 | 5 | -10 | 5 | 0 | x = 1 (repeated) |
| x² - 7x + 12 = 0 | 1 | -7 | 12 | 1 | x = 3, 4 |
| 3x² - 5x + 2 = 0 | 3 | -5 | 2 | 1 | x = 1, 0.667 |
| x² + 6x + 10 = 0 | 1 | 6 | 10 | -4 | x = -3 ± i |
Discriminant Analysis Chart
Δ > 0
Two distinct real roots
Parabola crosses x-axis twice
Δ = 0
One repeated real root
Parabola touches x-axis once
Δ < 0
Two complex roots
Parabola doesn't cross x-axis
Practice Problems
Problem 1:
Solve: x² - 5x + 6 = 0
Solution: Δ = 25 - 24 = 1, x = (5 ± 1)/2 = 3, 2
Problem 2:
Solve: 2x² + 3x - 2 = 0
Solution: Δ = 9 + 16 = 25, x = (-3 ± 5)/4 = 0.5, -2
Problem 3:
Solve: x² - 4x + 4 = 0
Solution: Δ = 16 - 16 = 0, x = 4/2 = 2 (repeated)
Problem 4:
Solve: x² + x + 1 = 0
Solution: Δ = 1 - 4 = -3, x = (-1 ± i√3)/2
Problem 5:
Solve: 3x² - 7x + 2 = 0
Solution: Δ = 49 - 24 = 25, x = (7 ± 5)/6 = 2, 1/3
Daily Uses of Quadratic Equations
Projectile motion calculations in physics and sports
Profit maximization problems in business economics
Area optimization in architecture and construction
Signal processing and electrical circuit analysis
Computer graphics and game development trajectories