1. **State the problem:** Solve the quadratic expression $3a^2 - 5a - 60 = 0$ for $a$. 2. **Formula and rules:** To solve a quadratic equation $ax^2 + bx + c = 0$, we can use the quadratic formula: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation. 3. **Identify coefficients:** Here, $a = 3$, $b = -5$, and $c = -60$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 3 \times (-60) = 25 + 720 = 745$$ 5. **Apply the quadratic formula:** $$a = \frac{-(-5) \pm \sqrt{745}}{2 \times 3} = \frac{5 \pm \sqrt{745}}{6}$$ 6. **Simplify the square root if possible:** $745 = 5 \times 149$, which has no perfect square factors, so leave as is. 7. **Final solutions:** $$a = \frac{5 + \sqrt{745}}{6} \quad \text{or} \quad a = \frac{5 - \sqrt{745}}{6}$$ These are the two solutions for $a$.