1. **State the problem:** Solve the quadratic equation $x^2 - 109 = -8x$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$x^2 + 8x - 109 = 0$$ 3. **Identify the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Apply the formula:** Here, $a=1$, $b=8$, and $c=-109$. Calculate the discriminant: $$b^2 - 4ac = 8^2 - 4 \times 1 \times (-109) = 64 + 436 = 500$$ 5. **Calculate the roots:** $$x = \frac{-8 \pm \sqrt{500}}{2}$$ Simplify $\sqrt{500}$: $$\sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5}$$ 6. **Write the exact solutions:** $$x = \frac{-8 \pm 10\sqrt{5}}{2}$$ 7. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{2}(-4 \pm 5\sqrt{5})}{\cancel{2}} = -4 \pm 5\sqrt{5}$$ **Final answer:** $$x = -4 + 5\sqrt{5} \quad \text{or} \quad x = -4 - 5\sqrt{5}$$