Solve the equation `x^(2)+(x/(x-1))^(2)=8`

Explanation

To solve the equation \( x^2 + \left( \frac{x}{x-1} \right)^2 = 8 \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ x^2 + \left( \frac{x}{x-1} \right)^2 = 8 \]

### Step 2: Substitute \( y = \frac{x}{x-1} \)
Let \( y = \frac{x}{x-1} \). Then, we can express \( x \) in terms of \( y \):
\[ x = y(x-1) \implies x = yx - y \implies x - yx = -y \implies x(1-y) = -y \implies x = \frac{-y}{1-y} \]

### Step 3: Substitute back into the equation
Now, substitute \( y \) back into the equation:
\[ x^2 + y^2 = 8 \]
Substituting \( y^2 \):
\[ x^2 + \left( \frac{x}{x-1} \right)^2 = 8 \]

### Step 4: Clear the fraction
Multiply through by \( (x-1)^2 \) to eliminate the denominator:
\[ x^2(x-1)^2 + x^2 = 8(x-1)^2 \]

### Step 5: Expand and simplify
Expand both sides:
\[ x^2(x^2 - 2x + 1) + x^2 = 8(x^2 - 2x + 1) \]
This simplifies to:
\[ x^4 - 2x^3 + x^2 + x^2 = 8x^2 - 16x + 8 \]
Combine like terms:
\[ x^4 - 2x^3 - 6x^2 + 16x - 8 = 0 \]

### Step 6: Factor or use the quadratic formula
We can try to factor or use the quadratic formula. However, let's assume we can find roots using substitution or numerical methods.

### Step 7: Solve the quadratic equation
Using the quadratic formula for \( x^2 - 4x + 4 = 0 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -4, c = 4 \):
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 16}}{2} = \frac{4 \pm 0}{2} = 2 \]

### Step 8: Solve for \( y = -2 \)
Now, we also have \( y^2 = -2 \):
\[ x^2 = -2(x-1) \implies x^2 + 2x - 2 = 0 \]
Using the quadratic formula again:
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 8}}{2} = \frac{-2 \pm \sqrt{12}}{2} = \frac{-2 \pm 2\sqrt{3}}{2} = -1 \pm \sqrt{3} \]

### Final Solutions
Thus, the solutions to the original equation are:
1. \( x = 2 \)
2. \( x = -1 + \sqrt{3} \)
3. \( x = -1 - \sqrt{3} \)