1. **Stating the problem:** Given the quadratic equation $$2y^2 - 5y - 3 = 0$$, find the roots \(\alpha\) and \(\beta\). 2. **Formula used:** For a quadratic equation $$ay^2 + by + c = 0$$, the roots are given by the quadratic formula: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 2$$, $$b = -5$$, and $$c = -3$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times (-3) = 25 + 24 = 49$$ 5. **Find the roots:** $$y = \frac{-(-5) \pm \sqrt{49}}{2 \times 2} = \frac{5 \pm 7}{4}$$ 6. **Evaluate each root:** - For the plus sign: $$y = \frac{5 + 7}{4} = \frac{12}{4} = 3$$ - For the minus sign: $$y = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$$ 7. **Final answer:** The roots are $$\alpha = 3$$ and $$\beta = -\frac{1}{2}$$.