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

Options

A. 16
B. `-16`
C. 8
D. -8

Answer

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

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

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

### Step 3: 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 already know \( \alpha + \beta = 2 \). Now we need to find \( \alpha^2 + \beta^2 \).

### Step 4: Calculate \( \alpha^2 + \beta^2 \)
Using the square of the sum of the roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the known values:
\[ \alpha^2 + \beta^2 = (2)^2 - 2 \cdot 4 = 4 - 8 = -4 \]

### Step 5: Substitute back into the identity
Now we can substitute \( \alpha^2 + \beta^2 \) into our equation for \( \alpha^3 + \beta^3 \):
\[ \alpha^3 + \beta^3 = (\alpha + \beta)((\alpha^2 + \beta^2) - \alpha\beta) \]
Substituting the values:
\[ \alpha^3 + \beta^3 = 2((-4) - 4) = 2(-8) = -16 \]

### Final Answer
Thus, the value of \( \alpha^3 + \beta^3 \) is \( -16 \).