If `x+(2)/(x)=1`, then the value of `(x^(2)+x+2)/(x^(2)(1-x))`
यदि `x+(2)/(x)=1` तो `(x^(2)+x+2)/(x^(2)(1-x))` का मान है:

Options

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

Explanation

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 \( x \)

Start with the equation:
\[ x + \frac{2}{x} = 1 \]

Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 + 2 = x \]

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 \( a = 1, b = -1, c = 2 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \]

Since the discriminant is negative, there are no real solutions for \( x \).

### Step 3: Substitute \( x \) in the expression

Since we have no real solutions, we can still analyze the expression \( \frac{x^2 + x + 2}{x^2(1 - x)} \).

From the original equation, we can express \( x^2 + 2 \) in terms of \( x \):
\[ x^2 + 2 = x \implies x^2 = x - 2 \]

### Step 4: Substitute \( x^2 \) into the expression

Now substitute \( x^2 \) in the expression \( \frac{x^2 + x + 2}{x^2(1 - x)} \):
\[ \frac{(x - 2) + x + 2}{(x - 2)(1 - x)} = \frac{2x}{(x - 2)(1 - x)} \]

### Step 5: Simplify the expression

Now simplify the expression:
\[ \frac{2x}{(x - 2)(1 - x)} \]

### Step 6: Analyze the denominator

The denominator can be rewritten:
\[ (x - 2)(1 - x) = -(x - 2)(x - 1) = -((x - 2)(x - 1)) \]

Thus, the expression becomes:
\[ \frac{2x}{-(x - 2)(x - 1)} = -\frac{2x}{(x - 2)(x - 1)} \]

### Step 7: Conclusion

Since we have no real solutions for \( x \) from the original equation, we cannot evaluate the expression for real values of \( x \). However, we can conclude that the expression simplifies to:
\[ -\frac{2x}{(x - 2)(x - 1)} \]