The number of solutions of ax² + bx + c = 0 is determined by the discriminant D = b² - 4ac.
The Discriminant and the Number of Solutions
- D > 0 → two distinct real solutions
- D = 0 → a repeated solution (one real solution)
- D < 0 → no real solutions (two imaginary solutions)
Example ①
Find k when x² - 4x + k = 0 has a repeated solution.
Solution:D = 16 - 4k = 0 → k = 4. The repeated solution is x = 2.
Find k when x² - 4x + k = 0 has a repeated solution.
Solution:D = 16 - 4k = 0 → k = 4. The repeated solution is x = 2.
Example ②
Find the range of k for which x² + 2x + k > 0 is always true.
Solution:D = 4 - 4k < 0 → k > 1
Find the range of k for which x² + 2x + k > 0 is always true.
Solution:D = 4 - 4k < 0 → k > 1
Key Points
- D = b² - 4ac (the discriminant)
- When D = 0, the repeated solution is x = -b/2a
- Conditions such as “always positive” and “always negative” are determined by the signs of a and D
Practice Problems
- Find the range of k for which x² - 6x + k = 0 has real solutions.
Solutions and Explanations
- Solution:k ≤ 9. D = 36 - 4k ≥ 0 → k ≤ 9.