If `alpha and beta` are roots of the equation `2x^(2)-3x-5=0`, then the value of `(1)/(alpha)+(1)/(beta)` is

Options

A. `-(3)/(5)`
B. `(3)/(5)`
C. `(5)/(3)`
D. `-(5)/(3)`

Answer

To solve the problem, we need to find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 2x^2 - 3x - 5 = 0 \).

### Step-by-Step Solution:

**Step 1: Identify coefficients from the quadratic equation.**

The given quadratic equation is:
\[ 2x^2 - 3x - 5 = 0 \]
Here, we can identify:
- \( a = 2 \)
- \( b = -3 \)
- \( c = -5 \)

**Step 2: Use Vieta's formulas to find \( \alpha + \beta \) and \( \alpha \beta \).**

According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-3}{2} = \frac{3}{2} \]

- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-5}{2} \]

**Step 3: Calculate \( \frac{1}{\alpha} + \frac{1}{\beta} \).**

We know that:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{\frac{3}{2}}{\frac{-5}{2}} \]

**Step 4: Simplify the expression.**

When we simplify:
\[ \frac{\frac{3}{2}}{\frac{-5}{2}} = \frac{3}{2} \times \frac{2}{-5} = \frac{3 \cdot 2}{2 \cdot -5} = \frac{3}{-5} = -\frac{3}{5} \]

### Final Answer:
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[ -\frac{3}{5} \]