How do you solve x² + 12x + 15 by completing the square, and what is the solution set of the equation?

To solve the quadratic equation x² + 12x + 15 = 0 by completing the square, we follow these steps:

1. Start with the equation:

x² + 12x + 15 = 0

2. Move the constant term to the right side:

x² + 12x = -15

3. To complete the square, take half of the coefficient of x, which is 12. Half of 12 is 6. Then, square it:

(6)² = 36

4. Add that square to both sides of the equation:

x² + 12x + 36 = -15 + 36

5. This simplifies to:

x² + 12x + 36 = 21

6. Now, the left side can be factored:

(x + 6)² = 21

7. Next, take the square root of both sides:

x + 6 = ±√21

8. Finally, solve for x by isolating it:

x = -6 ± √21

The solution set of the equation is:

x = -6 + √21 and x = -6 – √21

In conclusion, by completing the square, we have found the roots of the equation, which are expressed in terms of square roots.