To solve the problem of finding the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 3x^2 + 5x - 2 = 0 \), we can use some properties of roots of quadratic equations. ### Step 1: Understanding the Roots of a Quadratic Equation For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots \( \alpha \) and \( \beta \) can be found using Vieta's formulas: - The sum of the roots \( \alpha + \beta = -\frac{b}{a} \) - The product of the roots \( \alpha \beta = \frac{c}{a} \) In our case: - \( a = 3 \) - \( b = 5 \) - \( c = -2 \) ### Step 2: Calculate the Sum and Product of the Roots Using Vieta's formulas: 1. **Sum of the roots**: \[ \alpha + \beta = -\frac{b}{a} = -\frac{5}{3} \] 2. **Product of the roots**: \[ \alpha \beta = \frac{c}{a} = \frac{-2}{3} \] ### Step 3: Finding \( \frac{1}{\alpha} + \frac{1}{\beta} \) We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of the sum and product of the roots: \[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \] Substituting the values we found: \[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{5}{3}}{\frac{-2}{3}} \] ### Step 4: Simplifying the Expression Now, we simplify the fraction: \[ \frac{-\frac{5}{3}}{\frac{-2}{3}} = \frac{-5}{3} \times \frac{3}{-2} = \frac{5}{2} \] ### Conclusion Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is \( \frac{5}{2} \). ### Why the Other Options are Incorrect - **Option A: -5/3**: This value does not correspond to the sum or product of the roots and is not derived from the calculations. - **Option B: -2/3**: This is the product of the roots \( \alpha \beta \), not the sum of their reciprocals. - **Option C: 1/2**: This value does not relate to either the sum or product of the roots and is incorrect based on our calculations. ### Revision Summary - Use Vieta's formulas to find the sum and product of the roots of a quadratic equation. - The formula for the sum of the reciprocals of the roots is \( \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \). - Substitute the values of the sum and product into the formula to find the desired result. - Always check the options against your calculations to identify the correct answer. The correct answer is **D. \( \frac{5}{2} \)**.