Solve x² + 14x + 24 by Completing the Square. What is the Solution Set of the Equation?

To solve the equation x² + 14x + 24 by completing the square, follow these steps:

1. Start with the equation: x² + 14x + 24 = 0.
2. Move the constant term to the other side: x² + 14x = -24.
3. Next, to complete the square, take half of the coefficient of x (which is 14), square it, and add it to both sides:
• Half of 14 is 7, and squaring it gives us 49.
4. Add 49 to both sides: x² + 14x + 49 = -24 + 49.
5. This simplifies to (x + 7)² = 25.
6. Now, take the square root of both sides:
• x + 7 = ±5
7. Solving for x gives us two equations:
• x + 7 = 5 which simplifies to x = -2,
• x + 7 = -5 which simplifies to x = -12.
8. Thus, the solution set for the equation is {-2, -12}.

In conclusion, by completing the square, we find that the solutions are x = -2 and x = -12.