Sum of roots is -1 and sum of their reciprocals is `(1)/(6)`, then the equation is

Options

A. `x^(2)-x-6=0`
B. `x^(2)+x+6=0`
C. `x^(2)-x+6=0`
D. `x^(2)+x-6=0`

Answer

To solve the problem step by step, we need to find the quadratic equation given the sum of the roots and the sum of their reciprocals.

### Step 1: Define the roots
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \).

### Step 2: Use the given information
From the problem, we know:
- The sum of the roots:
\[ \alpha + \beta = -1 \]
- The sum of the reciprocals of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{6} \]

### Step 3: Simplify the sum of reciprocals
We can express the sum of the reciprocals in terms of the sum and product of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]
Substituting the known values:
\[ \frac{-1}{\alpha \beta} = \frac{1}{6} \]

### Step 4: Solve for the product of the roots
Cross-multiplying gives:
\[ -6 = \alpha \beta \]
Thus, we have:
\[ \alpha \beta = -6 \]

### Step 5: Form the quadratic equation
The standard form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we found:
\[ x^2 - (-1)x + (-6) = 0 \]
This simplifies to:
\[ x^2 + x - 6 = 0 \]

### Step 6: Write the final equation
Thus, the quadratic equation is:
\[ x^2 + x - 6 = 0 \]