If `alpha,beta` are the roots of the equation `x^2-2x+3=0` obtain the equation whose roots are `alpha^3-3alpha^2+5alpha-2` and `beta^3-beta^2+beta+5`

Options

A. `x^(2) = 3x + 2 = 0`
B. `x^(2) - 3x - 2 = 0`
C. `x^(2) - 3x + 2 = 0`
D. none of these

Explanation

To solve the problem, we need to find the equation whose roots are given by the expressions involving the roots \( \alpha \) and \( \beta \) of the quadratic equation \( x^2 - 2x + 3 = 0 \).

### Step 1: Identify the roots
The roots \( \alpha \) and \( \beta \) of the equation \( x^2 - 2x + 3 = 0 \) can be found using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -2, c = 3 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
Since the discriminant is negative, the roots are complex:
\[ \alpha, \beta = \frac{2 \pm \sqrt{-8}}{2} = 1 \pm i\sqrt{2} \]

### Step 2: Calculate the new roots
We need to find the new roots:
1. \( p = \alpha^3 - 3\alpha^2 + 5\alpha - 2 \)
2. \( q = \beta^3 - \beta^2 + \beta + 5 \)

#### For \( p \):
Using the equation \( \alpha^2 = 2\alpha - 3 \):
- Calculate \( \alpha^3 \):
\[ \alpha^3 = \alpha \cdot \alpha^2 = \alpha(2\alpha - 3) = 2\alpha^2 - 3\alpha \]
Substituting \( \alpha^2 \):
\[ \alpha^3 = 2(2\alpha - 3) - 3\alpha = 4\alpha - 6 - 3\alpha = \alpha - 6 \]
Now substitute into \( p \):
\[ p = (\alpha - 6) - 3(2\alpha - 3) + 5\alpha - 2 \]
Simplifying:
\[ p = \alpha - 6 - 6\alpha + 9 + 5\alpha - 2 = 0 + 1 = 1 \]

#### For \( q \):
Using the equation \( \beta^2 = 2\beta - 3 \):
- Calculate \( \beta^3 \):
\[ \beta^3 = \beta \cdot \beta^2 = \beta(2\beta - 3) = 2\beta^2 - 3\beta \]
Substituting \( \beta^2 \):
\[ \beta^3 = 2(2\beta - 3) - 3\beta = 4\beta - 6 - 3\beta = \beta - 6 \]
Now substitute into \( q \):
\[ q = (\beta - 6) - (2\beta - 3) + \beta + 5 \]
Simplifying:
\[ q = \beta - 6 - 2\beta + 3 + \beta + 5 = 0 + 2 = 2 \]

### Step 3: Form the quadratic equation
Now we have the roots \( p = 1 \) and \( q = 2 \). The quadratic equation with roots \( p \) and \( q \) can be formed using:
\[ x^2 - (p + q)x + pq = 0 \]
Substituting \( p \) and \( q \):
\[ x^2 - (1 + 2)x + (1 \cdot 2) = 0 \]
This simplifies to:
\[ x^2 - 3x + 2 = 0 \]

### Final Answer
The required equation is:
\[ \boxed{x^2 - 3x + 2 = 0} \]