1. **State the problem:** Solve the quadratic equation $$2v^2 + 11v + 13 = 0$$ for all real solutions in simplest form. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=11$, and $c=13$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 11^2 - 4 \times 2 \times 13 = 121 - 104 = 17$$ 4. **Apply the quadratic formula:** $$v = \frac{-11 \pm \sqrt{17}}{2 \times 2} = \frac{-11 \pm \sqrt{17}}{4}$$ 5. **Simplify the expression:** The solutions are $$v = \frac{-11 + \sqrt{17}}{4} \quad \text{and} \quad v = \frac{-11 - \sqrt{17}}{4}$$ These are the two real solutions in simplest radical form. **Final answer:** $$v = \frac{-11 \pm \sqrt{17}}{4}$$