If `c+ (1)/(x) = x + (1)/(c)` then the value of x

Options

A. `C ,-1//C`
B. `C, C ^(2)`
C. `C, 2 C`
D. `0,1`

Answer

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

### Step 1: Rearranging the equation
Start by rearranging the equation to isolate the terms involving \( x \) on one side and the constants on the other side.

\[ c + \frac{1}{x} - x = \frac{1}{c} \]

This simplifies to:

\[ c - x + \frac{1}{x} = \frac{1}{c} \]

### Step 2: Getting a common denominator
To eliminate the fractions, we can multiply through by \( cx \) (assuming \( x \neq 0 \) and \( c \neq 0 \)) to clear the denominators:

\[ c^2x - x^2c + 1 = x \]

### Step 3: Rearranging the equation
Rearranging gives us a standard quadratic form:

\[ cx^2 - c^2x + 1 = 0 \]

### Step 4: Applying the quadratic formula
Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = c \), \( b = -c^2 \), and \( c = 1 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-c^2)^2 - 4(c)(1) = c^4 - 4c \]

### Step 5: Finding the roots
Now substituting into the quadratic formula:

\[ x = \frac{-(-c^2) \pm \sqrt{c^4 - 4c}}{2c} \]

This simplifies to:

\[ x = \frac{c^2 \pm \sqrt{c^4 - 4c}}{2c} \]

### Step 6: Simplifying the expression
We can further simplify the expression:

\[ x = \frac{c}{2} \pm \frac{\sqrt{c^4 - 4c}}{2c} \]

### Conclusion
Thus, the values of \( x \) are:

\[ x = \frac{c}{2} + \frac{\sqrt{c^4 - 4c}}{2c} \quad \text{and} \quad x = \frac{c}{2} - \frac{\sqrt{c^4 - 4c}}{2c} \]