1. **State the problem:** Solve the quadratic equation $x^2 + 6x + 6 = 0$ using the square (completing the square) method. 2. **Formula and rules:** The square method involves rewriting the quadratic in the form $(x + p)^2 = q$ and then solving for $x$. This requires completing the square by adding and subtracting the same value. 3. **Step-by-step solution:** - Start with the equation: $$x^2 + 6x + 6 = 0$$ - Move the constant term to the right side: $$x^2 + 6x = -6$$ - Take half of the coefficient of $x$, which is $6$, half is $3$, and square it: $$3^2 = 9$$ - Add and subtract $9$ on the left side to complete the square: $$x^2 + 6x + 9 - 9 = -6$$ - Rewrite as a perfect square and simplify: $$(x + 3)^2 - 9 = -6$$ - Add $9$ to both sides: $$(x + 3)^2 = 3$$ - Take the square root of both sides: $$x + 3 = \pm \sqrt{3}$$ - Solve for $x$: $$x = -3 \pm \sqrt{3}$$ 4. **Final answer:** The solutions are $$x = -3 + \sqrt{3}$$ and $$x = -3 - \sqrt{3}$$.