All solutions of ax² + bx + c = 0（a ≠ 0） can be found using the quadratic formula.
x = (-b ± √(b² - 4ac)) / 2a
Example ①
Solve 3x² - 2x - 2 = 0 using the quadratic formula.
Solution:x = (2 ± √(4 + 24)) / 6 = (2 ± √28) / 6 = (1 ± √7) / 3
Solve 3x² - 2x - 2 = 0 using the quadratic formula.
Solution:x = (2 ± √(4 + 24)) / 6 = (2 ± √28) / 6 = (1 ± √7) / 3
Example ② (when b is even)
Solve x² - 6x + 4 = 0.
Solution:Let b' = -3. Then x = (6 ± √(36-16))/2 = 3 ± √5
Solve x² - 6x + 4 = 0.
Solution:Let b' = -3. Then x = (6 ± √(36-16))/2 = 3 ± √5
Key Points
- Identify a, b, and c before substituting
- When b is even: x = (-b' ± √(b'² - ac)) / a（b' = b/2）
- If the expression inside the radical is negative → no real solutions
Practice Problems
- Solve 2x² + 3x - 1 = 0 using the quadratic formula.
Solutions and Explanations
- Solution:x = (-3 ± √17) / 4. b²-4ac = 9+8=17.