We are given the following system of equations:

$\alpha = -2\alpha + k$ $\beta = -2\beta + k$

We need to determine the value of $\alpha^3 + \beta^3$.

Step 1: Solve for $\alpha$ and $\beta$

Let’s first solve for $\alpha$ and $\beta$ from the equations provided.

For $\alpha$:

$\alpha = -2\alpha + k$ Add $2\alpha$ to both sides: $\alpha + 2\alpha = k$ $3\alpha = k$ $\alpha = \frac{k}{3}$

For $\beta$:

$\beta = -2\beta + k$ Add $2\beta$ to both sides: $\beta + 2\beta = k$ $3\beta = k$ $\beta = \frac{k}{3}$

Step 2: Find $\alpha^3 + \beta^3$

Since $\alpha = \beta = \frac{k}{3}$, we have:

$\alpha^3 + \beta^3 = \left(\frac{k}{3}\right)^3 + \left(\frac{k}{3}\right)^3$

This simplifies to:

$\alpha^3 + \beta^3 = 2 \left(\frac{k^3}{27}\right) = \frac{2k^3}{27}$

Final Answer:

$\alpha^3 + \beta^3 = \frac{2k^3}{27}$

Would you like a more detailed explanation or further clarification?

Here are 5 related questions to think about:

1. How would the result change if $\alpha \neq \beta$?
2. What is the value of $\alpha^2 + \beta^2$?
3. Can this approach be generalized for any similar linear equation system?
4. How would the equation change if the coefficients of $\alpha$ and $\beta$ were different?
5. What is the interpretation of the result in terms of polynomial roots?

Tip: When dealing with equations involving variables, always solve the system step-by-step to isolate each variable.