1. **State the problem:** Solve the quadratic equation $$2x^2 + 9x - 5 = 0$$ 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, $a = 2$, $b = 9$, and $c = -5$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 9^2 - 4 \times 2 \times (-5) = 81 + 40 = 121$$ 5. **Apply the quadratic formula:** $$x = \frac{-9 \pm \sqrt{121}}{2 \times 2} = \frac{-9 \pm 11}{4}$$ 6. **Find the two roots:** - For the plus sign: $$x = \frac{-9 + 11}{4} = \frac{2}{4} = \frac{1}{2}$$ - For the minus sign: $$x = \frac{-9 - 11}{4} = \frac{-20}{4} = -5$$ 7. **Interpretation:** The roots are $x = \frac{1}{2}$ and $x = -5$. Among the multiple-choice answers, $\frac{1}{2}$ appears as options A and C, and $\frac{5}{2}$ is option E, but $-5$ is not listed. **Final answer:** The roots of the equation are $$x = \frac{1}{2} \text{ and } x = -5$$. The correct root from the options given is $\frac{1}{2}$.