How can I solve the equation x² + 14x + 24 by completing the square, and what is the solution set?

To solve the quadratic equation x² + 14x + 24 = 0 by completing the square, we will follow these steps:

1. Move the constant term to the other side:

   Start by rewriting the equation:

   x² + 14x = -24

2. Complete the square:

   To complete the square, we need to take the coefficient of x (which is 14), divide it by 2 (which gives us 7), and then square it (7² = 49).

   Add 49 to both sides of the equation:

   x² + 14x + 49 = -24 + 49

   Thus, the equation becomes:

   (x + 7)² = 25

3. Take the square root of both sides:

   Now, take the square root of both sides:

   x + 7 = ±5

4. Isolate x:

   Now, we’ll solve for x by isolating it for both scenarios:

   1. x + 7 = 5

   x = 5 – 7

   x = -2

   2. x + 7 = -5

   x = -5 – 7

   x = -12

The complete solution set for the equation x² + 14x + 24 = 0 is:

• x = -2
• x = -12

In conclusion, by completing the square, we find that the solution set is { -2, -12 }.