First make the coefficient of x² equal to 1, then transform the equation into the form (x + p)² = q by completing the square.
Example ①
Solve x² + 6x + 7 = 0 by completing the square.
Solution:(x + 3)² - 9 + 7 = 0 → (x + 3)² = 2 → x = -3 ± √2
Solve x² + 6x + 7 = 0 by completing the square.
Solution:(x + 3)² - 9 + 7 = 0 → (x + 3)² = 2 → x = -3 ± √2
Example ②
Solve 2x² - 4x - 1 = 0 by completing the square.
Solution:x² - 2x - 1/2 = 0 → (x - 1)² = 3/2 → x = 1 ± √6/2
Solve 2x² - 4x - 1 = 0 by completing the square.
Solution:x² - 2x - 1/2 = 0 → (x - 1)² = 3/2 → x = 1 ± √6/2
Key Points
- Add or subtract half the coefficient of x to make the form (x + p)²
- If the coefficient of x² is not 1, divide both sides
Practice Problems
- Solve x² - 4x + 1 = 0 by completing the square.
Solutions and Explanations
- Solution:x = 2 ± √3. (x-2)² = 3.