If `x + (2)/(x) = 3,` then the value of `(x ^(2) +x + 2)/(x ^(2) (3 - x))` is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( x + \frac{2}{x} = 3 \) and find the value of \( \frac{x^2 + x + 2}{x^2(3 - x)} \), we can follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x + \frac{2}{x} = 3 \]
Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 + 2 = 3x \]
Rearranging gives us:
\[ x^2 - 3x + 2 = 0 \]
This is a quadratic equation.

### Step 2: Factor the quadratic equation
We can factor the quadratic:
\[ (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 3: Substitute \( x \) back into the expression
Now we need to evaluate \( \frac{x^2 + x + 2}{x^2(3 - x)} \) for both values of \( x \).

#### For \( x = 1 \):
Substituting \( x = 1 \):
\[ \frac{1^2 + 1 + 2}{1^2(3 - 1)} = \frac{1 + 1 + 2}{1(2)} = \frac{4}{2} = 2 \]

#### For \( x = 2 \):
Substituting \( x = 2 \):
\[ \frac{2^2 + 2 + 2}{2^2(3 - 2)} = \frac{4 + 2 + 2}{4(1)} = \frac{8}{4} = 2 \]

### Conclusion
In both cases, the value of \( \frac{x^2 + x + 2}{x^2(3 - x)} \) is \( 2 \).

Thus, the final answer is:
\[ \boxed{2} \]