If `alpha and beta` are the roots of `x^(2) + 4 x + 6 = 0,` then what is the value of `alpha^(3) + beta^(3)` ?

Options

A. `-2//3`
B. `2//3`
C. 4
D. 8

Explanation

To find the value of \( \alpha^3 + \beta^3 \) where \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + 4x + 6 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is \( x^2 + 4x + 6 = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 6 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{4}{1} = -4 \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{6}{1} = 6 \)

### Step 3: Use the formula for \( \alpha^3 + \beta^3 \)
The formula for the sum of cubes of the roots is given by:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \]

### Step 4: Substitute the values
Now, substitute \( \alpha + \beta = -4 \) and \( \alpha \beta = 6 \) into the formula:
\[ \alpha^3 + \beta^3 = (-4)^3 - 3 \cdot 6 \cdot (-4) \]

### Step 5: Calculate \( (-4)^3 \)
Calculating \( (-4)^3 \):
\[ (-4)^3 = -64 \]

### Step 6: Calculate \( -3 \cdot 6 \cdot (-4) \)
Calculating \( -3 \cdot 6 \cdot (-4) \):
\[ -3 \cdot 6 \cdot (-4) = 72 \]

### Step 7: Combine the results
Now combine the results:
\[ \alpha^3 + \beta^3 = -64 + 72 = 8 \]

### Final Answer
Thus, the value of \( \alpha^3 + \beta^3 \) is \( 8 \).

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