If `alpha and beta` are the roots of the equation `x^(2)-x-4=0`, find the value of `(1)/(alpha)+(1)/(beta)-alpha beta`:

Options

A. `(16)/(7)`
B. `(8)/(5)`
C. `(15)/(4)`
D. none of these

Answer

To solve the problem, we need to find the value of \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta\) given that \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - x - 4 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given equation is \(x^2 - x - 4 = 0\). Here, we can identify:
- \(a = 1\)
- \(b = -1\)
- \(c = -4\)

2. **Use Vieta's formulas to find the sum and product of the roots**:
According to Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a} = -\frac{-1}{1} = 1\)
- The product of the roots \(\alpha \beta = \frac{c}{a} = \frac{-4}{1} = -4\)

3. **Calculate \(\frac{1}{\alpha} + \frac{1}{\beta}\)**:
We can express \(\frac{1}{\alpha} + \frac{1}{\beta}\) as:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{1}{-4} = -\frac{1}{4} \]

4. **Substitute into the expression**:
Now we substitute \(\frac{1}{\alpha} + \frac{1}{\beta}\) and \(\alpha \beta\) into the expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta = -\frac{1}{4} - (-4) \]
Simplifying this gives:
\[ -\frac{1}{4} + 4 = -\frac{1}{4} + \frac{16}{4} = \frac{15}{4} \]

5. **Final Answer**:
Therefore, the value of \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta\) is \(\frac{15}{4}\).

### Summary:
The value of \(\frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta\) is \(\frac{15}{4}\).