1. **State the problem:** Solve the equation $x^2 + 27 = -12x$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 + 12x + 27 = 0$$ 3. **Identify the quadratic form:** The equation is now in standard quadratic form $ax^2 + bx + c = 0$ where $a=1$, $b=12$, and $c=27$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 12^2 - 4 \times 1 \times 27 = 144 - 108 = 36$$ 6. **Find the roots:** $$x = \frac{-12 \pm \sqrt{36}}{2} = \frac{-12 \pm 6}{2}$$ 7. **Calculate each solution:** - For $+$ sign: $$x = \frac{-12 + 6}{2} = \frac{-6}{2} = -3$$ - For $-$ sign: $$x = \frac{-12 - 6}{2} = \frac{-18}{2} = -9$$ **Final answer:** The solutions are $x = -3$ and $x = -9$.