1. **State the problem:** Solve the quadratic equation $$3v^2 + 17v + 24 = 0$$ for $v$. 2. **Formula used:** The quadratic formula for solving $ax^2 + bx + c = 0$ is $$v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=17$, and $c=24$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 17^2 - 4 \times 3 \times 24 = 289 - 288 = 1$$ 4. **Apply the quadratic formula:** $$v = \frac{-17 \pm \sqrt{1}}{2 \times 3} = \frac{-17 \pm 1}{6}$$ 5. **Find the two solutions:** - For the plus sign: $$v = \frac{-17 + 1}{6} = \frac{-16}{6} = \frac{\cancel{-16}}{\cancel{6}} = -\frac{8}{3}$$ - For the minus sign: $$v = \frac{-17 - 1}{6} = \frac{-18}{6} = \frac{\cancel{-18}}{\cancel{6}} = -3$$ 6. **Final answer:** The solutions are $$v = -\frac{8}{3}$$ and $$v = -3$$.