If `x= (1)/(5-x) and x ne 5`, find `x^(3) + (1)/(x^(3))`

Answer

To solve the equation \( x = \frac{1}{5 - x} \) and find \( x^3 + \frac{1}{x^3} \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the equation:
\[ x = \frac{1}{5 - x} \]
Multiply both sides by \( 5 - x \) to eliminate the fraction:
\[ x(5 - x) = 1 \]

### Step 2: Expanding the Equation
Expanding the left side gives:
\[ 5x - x^2 = 1 \]

### Step 3: Rearranging into Standard Form
Rearranging the equation to standard quadratic form:
\[ -x^2 + 5x - 1 = 0 \]
Multiplying through by -1 gives:
\[ x^2 - 5x + 1 = 0 \]

### Step 4: Using the Quadratic Formula
To find \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -5 \), and \( c = 1 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{5 \pm \sqrt{25 - 4}}{2} \]
\[ x = \frac{5 \pm \sqrt{21}}{2} \]

### Step 5: Finding \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \frac{5 + \sqrt{21}}{2} + \frac{2}{5 + \sqrt{21}} \]
To simplify \( \frac{2}{5 + \sqrt{21}} \), multiply the numerator and denominator by \( 5 - \sqrt{21} \):
\[ \frac{2(5 - \sqrt{21})}{(5 + \sqrt{21})(5 - \sqrt{21})} = \frac{10 - 2\sqrt{21}}{25 - 21} = \frac{10 - 2\sqrt{21}}{4} = \frac{5 - \sqrt{21}}{2} \]
Thus:
\[ x + \frac{1}{x} = \frac{5 + \sqrt{21}}{2} + \frac{5 - \sqrt{21}}{2} = \frac{10}{2} = 5 \]

### Step 6: Finding \( x^2 + \frac{1}{x^2} \)
Using the identity:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \]
Substituting \( x + \frac{1}{x} = 5 \):
\[ x^2 + \frac{1}{x^2} = 5^2 - 2 = 25 - 2 = 23 \]

### Step 7: Finding \( x^3 + \frac{1}{x^3} \)
Using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 + \frac{1}{x^2}) - (x + \frac{1}{x}) \]
Substituting the known values:
\[ x^3 + \frac{1}{x^3} = 5 \cdot 23 - 5 = 115 - 5 = 110 \]

### Final Answer
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{110} \]