How do you solve x² + 6x + 7 by completing the square, and what is the solution set of the equation?

To solve the equation x² + 6x + 7 = 0 by completing the square, we can follow these steps:

1. Start with the equation: x² + 6x + 7 = 0.
2. Move the constant term to the other side: x² + 6x = -7.
3. To complete the square, we need to add and subtract the square of half the coefficient of x. The coefficient of x is 6, so half of this is 3, and its square is 9.
4. Add 9 to both sides of the equation:
• x² + 6x + 9 = -7 + 9
5. This simplifies to:
• (x + 3)² = 2
6. Now, take the square root of both sides. Remember to consider both the positive and negative roots:
• x + 3 = ±√2
7. Now, solve for x:
• x = -3 + √2
• x = -3 – √2

Thus, the solutions to the equation are:

• x = -3 + √2
• x = -3 – √2

In conclusion, the solution set of the equation x² + 6x + 7 = 0 is:

• { -3 + √2, -3 – √2 }