1. **State the problem:** Solve the quadratic equation $$x^2 + 2x - 35 = 0$$ 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:** $$x^2 + 2x - 35 = 0$$ Move the constant term to the right side: $$x^2 + 2x = 35$$ 4. **Complete the square:** Calculate $$\left(\frac{2}{2}\right)^2 = 1$$. Add and subtract 1 on the left side: $$x^2 + 2x + 1 - 1 = 35$$ Group the perfect square trinomial: $$\left(x + 1\right)^2 - 1 = 35$$ 5. **Isolate the perfect square:** $$\left(x + 1\right)^2 = 35 + 1$$ $$\left(x + 1\right)^2 = 36$$ 6. **Take the square root of both sides:** $$x + 1 = \pm \sqrt{36}$$ $$x + 1 = \pm 6$$ 7. **Solve for $$x$$:** For the positive root: $$x + 1 = 6$$ $$x = 6 - 1$$ $$x = 5$$ For the negative root: $$x + 1 = -6$$ $$x = -6 - 1$$ $$x = -7$$ **Final answer:** $$x = 5$$ or $$x = -7$$