The roots of the equation `x/(x-1)+(x-1)/x=2 1/2` are

Options

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

Answer

To solve the equation \( \frac{x}{x-1} + \frac{x-1}{x} = 2 \frac{1}{2} \), we will follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The mixed fraction \( 2 \frac{1}{2} \) can be converted to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \]

### Step 2: Rewrite the equation
Now, we can rewrite the equation as:
\[ \frac{x}{x-1} + \frac{x-1}{x} = \frac{5}{2} \]

### Step 3: Cross-multiply
Cross-multiplying gives:
\[ 2 \left( \frac{x}{x-1} + \frac{x-1}{x} \right) = 5 \]
This leads to:
\[ 2x + 2 \frac{(x-1)^2}{x} = 5(x-1) \]

### Step 4: Clear the fractions
Multiply through by \( x(x-1) \) to eliminate the denominators:
\[ 2x^2 + 2(x-1)^2 = 5x(x-1) \]

### Step 5: Expand the terms
Expanding both sides:
- Left side:
\[ 2x^2 + 2(x^2 - 2x + 1) = 2x^2 + 2x^2 - 4x + 2 = 4x^2 - 4x + 2 \]
- Right side:
\[ 5x^2 - 5x \]

### Step 6: Set the equation to zero
Now we have:
\[ 4x^2 - 4x + 2 = 5x^2 - 5x \]
Rearranging gives:
\[ 4x^2 - 4x + 2 - 5x^2 + 5x = 0 \]
This simplifies to:
\[ -x^2 + x + 2 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - x - 2 = 0 \]

### Step 7: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ x^2 - x - 2 = (x - 2)(x + 1) = 0 \]

### Step 8: Find the roots
Setting each factor to zero gives us the roots:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Final Answer
The roots of the equation are \( x = 2 \) and \( x = -1 \).

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