1. **State the problem:** Solve the quadratic equation $$x^2 - 12x + 35 = 0$$ by completing the square. 2. **Recall the formula and rule:** To complete the square for an equation of the form $$x^2 + bx + c = 0$$, we rewrite it as $$\left(x - \frac{b}{2}\right)^2 = \text{some number}$$ by adding and subtracting $$\left(\frac{b}{2}\right)^2$$. 3. **Rewrite the equation:** $$x^2 - 12x + 35 = 0$$ Move the constant term to the right side: $$x^2 - 12x = -35$$ 4. **Complete the square:** Calculate $$\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36$$. Add 36 to both sides: $$x^2 - 12x + 36 = -35 + 36$$ Show cancellation: $$x^2 - 12x + \cancel{36} = -35 + \cancel{36}$$ 5. **Rewrite left side as a perfect square:** $$\left(x - 6\right)^2 = 1$$ 6. **Solve for x:** Take the square root of both sides: $$x - 6 = \pm \sqrt{1}$$ $$x - 6 = \pm 1$$ 7. **Find the two solutions:** $$x = 6 \pm 1$$ So, $$x = 6 + 1 = 7$$ $$x = 6 - 1 = 5$$ **Final answer:** $$\boxed{x = 5 \text{ or } x = 7}$$