1. **State the problem:** Solve the equation $$4x - 9 + 9x + 6 = 7x^2 - 3x - 54$$ for $x$. 2. **Combine like terms on the left side:** $$4x - 9 + 9x + 6 = (4x + 9x) + (-9 + 6) = 13x - 3$$ So the equation becomes: $$13x - 3 = 7x^2 - 3x - 54$$ 3. **Bring all terms to one side to set the equation to zero:** $$0 = 7x^2 - 3x - 54 - 13x + 3$$ Simplify the right side: $$0 = 7x^2 - 16x - 51$$ 4. **Use the quadratic formula to solve for $x$:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=7$, $b=-16$, and $c=-51$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-16)^2 - 4 \times 7 \times (-51) = 256 + 1428 = 1684$$ 6. **Calculate the square root of the discriminant:** $$\sqrt{1684} = 2\sqrt{421}$$ (since $1684 = 4 \times 421$) 7. **Write the solutions:** $$x = \frac{-(-16) \pm 2\sqrt{421}}{2 \times 7} = \frac{16 \pm 2\sqrt{421}}{14}$$ 8. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2}(8 \pm \sqrt{421})}{\cancel{2}7} = \frac{8 \pm \sqrt{421}}{7}$$ 9. **Approximate the solutions:** $$\sqrt{421} \approx 20.518$$ So, $$x_1 = \frac{8 + 20.518}{7} = \frac{28.518}{7} \approx 4.074$$ $$x_2 = \frac{8 - 20.518}{7} = \frac{-12.518}{7} \approx -1.788$$ 10. **Check which option matches:** Options given are: - Option 2: $\frac{28}{3} \approx 9.333$ (no) - Option 3: $\frac{11}{2} = 5.5$ (no) - Option 4: $\frac{64}{13} \approx 4.923$ (closest to $4.074$ but not exact) - Option 1: $\emptyset$ (no solution) None of the options exactly match the solutions, so the solution set is: $$\left\{ \frac{8 + \sqrt{421}}{7}, \frac{8 - \sqrt{421}}{7} \right\}$$