Let `alpha`, `beta` be two real numbers satisfying the following relations `alpha^(2)+beta^(2)=5`, `3(alpha^(5)+beta^(5))=11(alpha^(3)+beta^(3))`
Quadratic equation having roots `alpha` and `beta` is

Options

A. `x^(2)+-x+2=0`
B. `x^(2)+-3x-2=0`
C. `x^(2)+-sqrt(3)x+2=0`
D. none of these

Explanation

To find the quadratic equation having roots \(\alpha\) and \(\beta\) given the conditions \(\alpha^2 + \beta^2 = 5\) and \(3(\alpha^5 + \beta^5) = 11(\alpha^3 + \beta^3)\), we can follow these steps:

### Step 1: Use the identity for \(\alpha^5 + \beta^5\)

We can express \(\alpha^5 + \beta^5\) in terms of \(\alpha^3 + \beta^3\) and \(\alpha^2 + \beta^2\). The identity is:

\[ \alpha^5 + \beta^5 = (\alpha + \beta)(\alpha^4 + \beta^4) - \alpha\beta(\alpha^3 + \beta^3) \]

We also know that:

\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2\alpha^2\beta^2 \]

### Step 2: Express \(\alpha^3 + \beta^3\)

Using the identity:

\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) \]

### Step 3: Substitute and simplify

From the given equations, we can substitute the expressions we derived:

1. Substitute \(\alpha^2 + \beta^2 = 5\).
2. Substitute \(\alpha^3 + \beta^3\) into the equation \(3(\alpha^5 + \beta^5) = 11(\alpha^3 + \beta^3)\).

This gives us:

\[ 3\left((\alpha + \beta)(\alpha^4 + \beta^4) - \alpha\beta(\alpha^3 + \beta^3)\right) = 11(\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) \]

### Step 4: Substitute \(\alpha^4 + \beta^4\)

Using the expression for \(\alpha^4 + \beta^4\):

\[ \alpha^4 + \beta^4 = 5^2 - 2\alpha^2\beta^2 = 25 - 2\alpha^2\beta^2 \]

### Step 5: Rearranging and solving

Substituting everything back into the equation and simplifying will yield a polynomial in terms of \(\alpha + \beta\) and \(\alpha\beta\).

Let \(s = \alpha + \beta\) and \(p = \alpha\beta\). The equation simplifies to:

\[ 3s(25 - 2p) - 3p\left(s\left(5 - p\right)\right) = 11s(5 - p) \]

### Step 6: Solve for \(p\)

After simplifying, we can isolate \(p\) and solve for its value.

### Step 7: Find \(s\)

Using the relation \(\alpha^2 + \beta^2 = s^2 - 2p = 5\), we can find \(s\) based on the value of \(p\).

### Step 8: Form the quadratic equation

The quadratic equation with roots \(\alpha\) and \(\beta\) is given by:

\[ x^2 - sx + p = 0 \]

### Final Result

After calculating \(s\) and \(p\), we can write the final quadratic equation.