1. **State the problem:** Solve the quadratic equation $$x^2 - 39 = -2x + 9$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 39 + 2x - 9 = 0$$ 3. **Simplify:** Combine like terms: $$x^2 + 2x - 48 = 0$$ 4. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a=1$$, $$b=2$$, and $$c=-48$$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times (-48) = 4 + 192 = 196$$ 6. **Find the square root of the discriminant:** $$\sqrt{196} = 14$$ 7. **Calculate the two solutions:** $$x = \frac{-2 \pm 14}{2 \times 1} = \frac{-2 \pm 14}{2}$$ 8. **First solution:** $$x = \frac{-2 + 14}{2} = \frac{12}{2} = 6$$ 9. **Second solution:** $$x = \frac{-2 - 14}{2} = \frac{-16}{2} = -8$$ **Final answer:** The solutions to the equation are $$x = 6$$ and $$x = -8$$.