1. **State the problem:** Find $x$ if $9x^2 + 3x + 1 = 28$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$9x^2 + 3x + 1 - 28 = 0$$ which simplifies to $$9x^2 + 3x - 27 = 0$$ 3. **Identify the quadratic equation:** It is in the form $ax^2 + bx + c = 0$ where $a=9$, $b=3$, and $c=-27$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 9 \times (-27) = 9 + 972 = 981$$ 6. **Find the square root of the discriminant:** $$\sqrt{981} = \sqrt{9 \times 109} = 3\sqrt{109}$$ 7. **Substitute values into the quadratic formula:** $$x = \frac{-3 \pm 3\sqrt{109}}{2 \times 9} = \frac{-3 \pm 3\sqrt{109}}{18}$$ 8. **Simplify the expression:** $$x = \frac{3(-1 \pm \sqrt{109})}{18} = \frac{-1 \pm \sqrt{109}}{6}$$ **Final answer:** $$x = \frac{-1 + \sqrt{109}}{6} \quad \text{or} \quad x = \frac{-1 - \sqrt{109}}{6}$$