If `alpha+beta=3, alpha^3+beta^3=7`, then `alpha` and `beta` are the roots of

Options

A. `3x^2+9x+7=0`
B. `9x^2-27x+20=0`
C. `2x^2-6x+15=0`
D. None of these

Answer

To find the quadratic equation whose roots are \( \alpha \) and \( \beta \), we can follow these steps:

### Step 1: Identify the given information
We are given:
- \( \alpha + \beta = 3 \)
- \( \alpha^3 + \beta^3 = 7 \)

### Step 2: Use the identity for the sum of cubes
We can use the identity for the sum of cubes:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) \]
We can also express \( \alpha^2 + \beta^2 \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Thus, we can rewrite \( \alpha^3 + \beta^3 \) as:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)((\alpha + \beta)^2 - 3\alpha\beta) \]

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ 7 = 3 \left(3^2 - 3\alpha\beta\right) \]
This simplifies to:
\[ 7 = 3 \left(9 - 3\alpha\beta\right) \]

### Step 4: Solve for \( \alpha\beta \)
Expanding the equation:
\[ 7 = 27 - 9\alpha\beta \]
Rearranging gives:
\[ 9\alpha\beta = 27 - 7 \]
\[ 9\alpha\beta = 20 \]
\[ \alpha\beta = \frac{20}{9} \]

### Step 5: Form the quadratic equation
Now we can form the quadratic equation using the sum and product of the roots:
\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \]
Substituting the values we found:
\[ x^2 - 3x + \frac{20}{9} = 0 \]

### Step 6: Clear the fraction
To eliminate the fraction, multiply the entire equation by 9:
\[ 9x^2 - 27x + 20 = 0 \]

### Conclusion
Thus, the quadratic equation whose roots are \( \alpha \) and \( \beta \) is:
\[ 9x^2 - 27x + 20 = 0 \]