algebra
The Quadratic Formula
⚡ THE FORMULA
x = (-b ± √(b² - 4ac)) / 2a
"Negative b, plus or minus the square root of b squared minus 4ac, all over 2a"
Step-by-Step Example
Let's solve x² - 5x + 6 = 0
1
Identify a, b, and c
a = 1, b = -5, c = 62
Calculate the discriminant
b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 13
Take the square root
√1 = 14
Apply the formula
x = (5 ± 1) / 25
Calculate both solutions
x₁ = (5 + 1)/2 = 3, x₂ = (5 - 1)/2 = 2Solutions: x = 3 and x = 2✓
💡When to Use the Quadratic Formula
- When the equation doesn't factor easily
- When you need exact answers (not estimates)
- When dealing with complex roots
- Always works - it's the universal method!
More Examples
2x² + 3x - 2 = 0x = 0.5, x = -2a=2, b=3, c=-2 → D=25 → x = (-3±5)/4
x² - 4 = 0x = 2, x = -2a=1, b=0, c=-4 → D=16 → x = (0±4)/2
x² + 4x + 4 = 0x = -2 (repeated)a=1, b=4, c=4 → D=0 → x = -4/2
Try It Yourself
Enter your coefficients:
x² +x += 0
⚠️ Common Mistakes
- Forgetting the ± (there are TWO solutions!)
- Wrong sign for b (if b = -5, then -b = 5)
- Forgetting to divide the ENTIRE numerator by 2a
- Calculation errors in b² - 4ac