If `x + (2)/(x) =1,` then the value fo `(x ^(2) + x + 2)/( x ^(2) (1-x))` is :

Options

A. `-1`
B. 2
C. `-2`
D. 1

Answer

To solve the equation \( x + \frac{2}{x} = 1 \) and find the value of \( \frac{x^2 + x + 2}{x^2(1-x)} \), we can follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x + \frac{2}{x} = 1 \]
Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 2 = x \]
Rearranging gives:
\[ x^2 - x + 2 = 0 \]

### Step 2: Use the quadratic formula
To find the roots of the quadratic equation \( x^2 - x + 2 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = 2 \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 - 8}}{2} \]
\[ x = \frac{1 \pm \sqrt{-7}}{2} \]
This indicates that \( x \) has complex solutions:
\[ x = \frac{1 \pm i\sqrt{7}}{2} \]

### Step 3: Substitute \( x \) into the expression
Now we need to evaluate:
\[ \frac{x^2 + x + 2}{x^2(1-x)} \]
First, we need to find \( x^2 \):
Using \( x = \frac{1 + i\sqrt{7}}{2} \):
\[ x^2 = \left(\frac{1 + i\sqrt{7}}{2}\right)^2 = \frac{(1 + i\sqrt{7})^2}{4} = \frac{1 + 2i\sqrt{7} - 7}{4} = \frac{-6 + 2i\sqrt{7}}{4} = \frac{-3 + i\sqrt{7}}{2} \]

### Step 4: Calculate \( x^2 + x + 2 \)
Now we calculate \( x^2 + x + 2 \):
\[ x^2 + x + 2 = \frac{-3 + i\sqrt{7}}{2} + \frac{1 + i\sqrt{7}}{2} + 2 \]
Combining the terms:
\[ = \frac{-3 + 1 + 2 + 2i\sqrt{7}}{2} = \frac{0 + 2i\sqrt{7}}{2} = i\sqrt{7} \]

### Step 5: Calculate \( 1 - x \)
Now calculate \( 1 - x \):
\[ 1 - x = 1 - \frac{1 + i\sqrt{7}}{2} = \frac{2 - (1 + i\sqrt{7})}{2} = \frac{1 - i\sqrt{7}}{2} \]

### Step 6: Substitute into the expression
Now substitute \( x^2 \) and \( 1 - x \) into the expression:
\[ \frac{i\sqrt{7}}{\left(\frac{-3 + i\sqrt{7}}{2}\right) \left(\frac{1 - i\sqrt{7}}{2}\right)} \]
Calculating the denominator:
\[ = \frac{-3(1) + 3(i\sqrt{7}) + (-i\sqrt{7})(1) + (i\sqrt{7})(-i\sqrt{7})}{4} = \frac{-3 + 2i\sqrt{7} + 7}{4} = \frac{4 + 2i\sqrt{7}}{4} = 1 + \frac{i\sqrt{7}}{2} \]

### Final Expression
Thus, the overall expression simplifies to:
\[ \frac{i\sqrt{7}}{1 + \frac{i\sqrt{7}}{2}} = 1 \]

### Conclusion
The value of \( \frac{x^2 + x + 2}{x^2(1-x)} \) is \( 1 \).