1. **State the problem:** Solve the quadratic equation $$3x^2 - 7x - 288 = 0$$ for $x$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 3$, $b = -7$, and $c = -288$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 3 \times (-288) = 49 + 3456 = 3505$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-7) \pm \sqrt{3505}}{2 \times 3} = \frac{7 \pm \sqrt{3505}}{6}$$ 6. **Simplify the square root if possible:** 3505 factors as $3505 = 5 \times 701$, and 701 is prime, so no perfect square factors other than 1. 7. **Final solutions:** $$x = \frac{7 + \sqrt{3505}}{6} \quad \text{or} \quad x = \frac{7 - \sqrt{3505}}{6}$$ These are the exact solutions to the quadratic equation.