Question

Solve x²−8x−24=0 by completing the square. If there are multiple solutions, simplify the solution and enter them from least to greatest, separated by a comma.

Answer 1

The quadratic equation x² - 8x - 24 = 0 is solved by completing the square, leading to the solutions x = 4 - 2√10 and x = 4 + 2√10 when ordered from least to greatest.

Step-by-step explanation:

To solve the quadratic equation x² - 8x - 24 = 0 by completing the square, follow these steps:

1. Move the constant term to the other side of the equation: x² - 8x = 24.
2. Find the number that completes the square for the x² - 8x term. This is done by taking half of the coefficient of x (which is -8) and squaring it (4² = 16).
3. Add 16 to both sides of the equation to maintain equality: x² - 8x + 16 = 24 + 16.
4. The left side of the equation now forms a perfect square: (x - 4)² = 40.
5. Take the square root of both sides: x - 4 = ±√40.
6. Solve for x by adding 4 to both possible square roots: x = 4 ± √40.
7. Simplify the square root of 40 to get the final solutions: x = 4 ± 2√10. So, the solutions are x = 4 + 2√10 and x = 4 - 2√10.

Therefore, the solutions to the equation x² - 8x - 24 = 0 are x = 4 - 2√10 and x = 4 + 2√10, in that order when listed from least to greatest.