If `alpha and beta` are the zeroes of `x^2 - 8x + 1`, then the value of `1/alpha +1/beta - alpha beta` is :

Options

A. 7
B. 1
C. 5
D. `-8`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 8x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is \( x^2 - 8x + 1 \). Here, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -8 \) (coefficient of \( x \))
- \( c = 1 \) (constant term)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Calculating these:
- \( \alpha + \beta = -\frac{-8}{1} = 8 \)
- \( \alpha \beta = \frac{1}{1} = 1 \)

### Step 3: Calculate \( \frac{1}{\alpha} + \frac{1}{\beta} \)
Using the formula for the sum of reciprocals of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]
Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{8}{1} = 8 \]

### Step 4: Substitute into the expression
Now we substitute \( \frac{1}{\alpha} + \frac{1}{\beta} \) and \( \alpha \beta \) into the expression \( \frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta \):
\[ \frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta = 8 - 1 \]

### Step 5: Calculate the final result
Now, performing the subtraction:
\[ 8 - 1 = 7 \]

Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} - \alpha \beta \) is \( \boxed{7} \).