1. **State the problem:** Solve the quadratic equation $x^2 - 2x = 8$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^2 - 2x - 8 = 0$$ 3. **Identify the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-2$, and $c=-8$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36$$ 5. **Find the square root of the discriminant:** $$\sqrt{36} = 6$$ 6. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm 6}{2(1)} = \frac{2 \pm 6}{2}$$ 7. **Calculate the two solutions:** - For the plus sign: $$x = \frac{2 + 6}{2} = \frac{8}{2} = 4$$ - For the minus sign: $$x = \frac{2 - 6}{2} = \frac{\cancel{2} - 6}{\cancel{2}} = \frac{-4}{2} = -2$$ 8. **Final answer:** The solutions to the equation are: $$x = 4 \quad \text{or} \quad x = -2$$