If `alpha and beta are the roots of the quadratic equation `x^(2) + px + q = 0`, then form the quadratic equation whose roots are `alpha + (1)/(beta), beta + (1)/(alpha)`

Answer

To find the quadratic equation whose roots are \( \alpha + \frac{1}{\beta} \) and \( \beta + \frac{1}{\alpha} \), we start with the given quadratic equation:

\[ x^2 + px + q = 0 \]

where \( \alpha \) and \( \beta \) are the roots. From Vieta's formulas, we know:

1. The sum of the roots \( \alpha + \beta = -p \)
2. The product of the roots \( \alpha \beta = q \)

### Step 1: Find the new roots

The new roots are given as:

\[ r_1 = \alpha + \frac{1}{\beta} \]
\[ r_2 = \beta + \frac{1}{\alpha} \]

### Step 2: Simplify the new roots

We can rewrite \( r_1 \) and \( r_2 \):

\[ r_1 = \alpha + \frac{1}{\beta} = \alpha + \frac{\alpha}{\alpha \beta} = \frac{\alpha^2 + 1}{\beta} \]
\[ r_2 = \beta + \frac{1}{\alpha} = \beta + \frac{\beta}{\alpha \beta} = \frac{\beta^2 + 1}{\alpha} \]

### Step 3: Find the sum of the new roots

Now, we calculate the sum of the new roots:

\[ r_1 + r_2 = \left( \alpha + \frac{1}{\beta} \right) + \left( \beta + \frac{1}{\alpha} \right) = \alpha + \beta + \frac{1}{\beta} + \frac{1}{\alpha} \]

Using \( \alpha + \beta = -p \) and \( \frac{1}{\beta} + \frac{1}{\alpha} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-p}{q} \):

\[ r_1 + r_2 = -p + \frac{-p}{q} = -p\left(1 + \frac{1}{q}\right) = -p \cdot \frac{q + 1}{q} \]

### Step 4: Find the product of the new roots

Next, we calculate the product of the new roots:

\[ r_1 \cdot r_2 = \left( \alpha + \frac{1}{\beta} \right) \left( \beta + \frac{1}{\alpha} \right) \]
\[ = \alpha \beta + \alpha \cdot \frac{1}{\alpha} + \beta \cdot \frac{1}{\beta} + \frac{1}{\alpha \beta} \]
\[ = \alpha \beta + 1 + 1 + \frac{1}{\alpha \beta} = q + 2 + \frac{1}{q} \]

### Step 5: Form the new quadratic equation

The quadratic equation with roots \( r_1 \) and \( r_2 \) can be expressed as:

\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]

Substituting the values we found:

\[ x^2 + p \cdot \frac{q + 1}{q} x - \left(q + 2 + \frac{1}{q}\right) = 0 \]

### Final Result

Thus, the required quadratic equation is:

\[ x^2 + \frac{p(q + 1)}{q} x - \left(q + 2 + \frac{1}{q}\right) = 0 \]