Given `alpha and beta` are the roots of the quadratic equation `x^(2)-4x+k=0(kne0).` If `alpha beta, alpha beta^(2)+alpha^(2)beta and alpha^(3)+beta^(3)` are in geometric progression, then the value of k is equal to

Options

A. 4
B. `(16)/(7)`
C. `(3)/(7)`
D. 12

Answer

To solve the problem step by step, we will follow the mathematical reasoning based on the given conditions.

### Step 1: Identify the roots and their relationships
Given the quadratic equation \( x^2 - 4x + k = 0 \), we know that:
- The sum of the roots \( \alpha + \beta = 4 \)
- The product of the roots \( \alpha \beta = k \)