Let p and q be real numbers such that `p ne 0, p^(3) ne q and p^(3) ne -q.` If `alpha and beta` are non-zero complex number satisfying `alpha +beta=-p and alpha^(3)+beta^(3)=q`, then a quadratic equation having `(alpha)/(beta) ,(beta)/(alpha)` as its root is

Options

A. `(p^(3)+q)x^(2)-(p^(3)+2q)x+(p^(3)+q)=0`
B. `(p^(3)+q)x^(2)-(p^(3)-2q)x+(p^(3)+q)=0`
C. `(p^(3)-q) x^(2)-(5p^(3)-2q)x+(p^(3)-q)=0`
D. `(p^(3)-q)x^(2)-(5p^(3)+2q)x+(p^(3)-q)=0`

Answer

To solve the problem, we need to find a quadratic equation with roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). We will follow these steps:

### Step 1: Identify the Roots
The roots of the quadratic equation we want to form are:
\[ r_1 = \frac{\alpha}{\beta}, \quad r_2 = \frac{\beta}{\alpha} \]

### Step 2: Calculate the Sum of Roots
The sum of the roots \(r_1 + r_2\) can be calculated as follows:
\[ r_1 + r_2 = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

### Step 3: Calculate the Product of Roots
The product of the roots \(r_1 \cdot r_2\) is:
\[ r_1 \cdot r_2 = \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 4: Form the Quadratic Equation
Using the standard form of a quadratic equation \(x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0\), we substitute the values we found:
\[ x^2 - \left(\frac{\alpha^2 + \beta^2}{\alpha \beta}\right)x + 1 = 0 \]

### Step 5: Find \(\alpha^2 + \beta^2\)
We know from the problem that:
\[ \alpha + \beta = -p \]
Squaring both sides gives:
\[ (\alpha + \beta)^2 = p^2 \implies \alpha^2 + \beta^2 + 2\alpha\beta = p^2 \implies \alpha^2 + \beta^2 = p^2 - 2\alpha\beta \]

### Step 6: Substitute \(\alpha^2 + \beta^2\) into the Equation
Now, we need to express \(\alpha \beta\). From the identity for \(\alpha^3 + \beta^3\):
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) = q \]
Substituting \(\alpha + \beta = -p\):
\[ q = -p(\alpha^2 - \alpha\beta + \beta^2) = -p((\alpha^2 + \beta^2) - \alpha\beta) \]
Substituting \(\alpha^2 + \beta^2 = p^2 - 2\alpha\beta\):
\[ q = -p((p^2 - 2\alpha\beta) - \alpha\beta) = -p(p^2 - 3\alpha\beta) \]
Rearranging gives:
\[ \alpha\beta = \frac{p^3 + q}{3p} \]

### Step 7: Substitute \(\alpha\beta\) into the Quadratic Equation
Now substituting \(\alpha^2 + \beta^2\) and \(\alpha\beta\) into the quadratic equation:
\[ x^2 - \left(\frac{p^2 - 2\left(\frac{p^3 + q}{3p}\right)}{\frac{p^3 + q}{3p}}\right)x + 1 = 0 \]

### Final Form of the Quadratic Equation
After simplification, we multiply through by \(3p\) to eliminate the fraction:
\[ 3p x^2 - (p^3 + q - 2q)x + 3p = 0 \]
This simplifies to:
\[ 3p x^2 - (p^3 - q)x + 3p = 0 \]

### Conclusion
The quadratic equation having roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is:
\[ 3p x^2 - (p^3 - q)x + 3p = 0 \]