1. **Problem statement:** Solve the quadratic equation $x^2 - 6x - 3 = 0$ by completing the square. 2. **Recall the method:** To complete the square for an equation of the form $x^2 + bx + c = 0$, we move the constant term to the right side and add the square of half the coefficient of $x$ to both sides. 3. **Step-by-step solution:** - Start with the equation: $$x^2 - 6x - 3 = 0$$ - Move the constant term to the right: $$x^2 - 6x = 3$$ - Take half of the coefficient of $x$, which is $-6$, half is $-3$, and square it: $$(-3)^2 = 9$$ - Add $9$ to both sides to complete the square: $$x^2 - 6x + 9 = 3 + 9$$ - This forms a perfect square trinomial on the left: $$(x - 3)^2 = 12$$ - Take the square root of both sides: $$x - 3 = \pm \sqrt{12}$$ - Simplify the square root: $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ - Solve for $x$: $$x = 3 \pm 2\sqrt{3}$$ 4. **Final answer:** $$x = 3 \pm 2\sqrt{3}$$ This completes the solution by completing the square.