The solution set of ` (x+(1)/(x) ) ^2 -3/2 (x-(1)/(x))` =4 when `x ne 0` is

Answer

To solve the equation \( (x + \frac{1}{x})^2 - \frac{3}{2}(x - \frac{1}{x}) = 4 \) where \( x \neq 0 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (x + \frac{1}{x})^2 - \frac{3}{2}(x - \frac{1}{x}) = 4 \]

### Step 2: Use the identity for squares
We know that:
\[ (x + \frac{1}{x})^2 = (x - \frac{1}{x})^2 + 4 \]
This means we can rewrite the left side of our equation:
\[ (x - \frac{1}{x})^2 + 4 - \frac{3}{2}(x - \frac{1}{x}) = 4 \]

### Step 3: Simplify the equation
Subtracting 4 from both sides gives:
\[ (x - \frac{1}{x})^2 - \frac{3}{2}(x - \frac{1}{x}) = 0 \]
Let \( t = x - \frac{1}{x} \). Then, we can rewrite the equation as:
\[ t^2 - \frac{3}{2}t = 0 \]

### Step 4: Factor the equation
Factoring out \( t \) gives:
\[ t(t - \frac{3}{2}) = 0 \]
This gives us two possible solutions for \( t \):
1. \( t = 0 \)
2. \( t = \frac{3}{2} \)

### Step 5: Solve for \( x \) when \( t = 0 \)
If \( t = 0 \):
\[ x - \frac{1}{x} = 0 \implies x^2 - 1 = 0 \implies (x - 1)(x + 1) = 0 \]
Thus, \( x = 1 \) or \( x = -1 \).

### Step 6: Solve for \( x \) when \( t = \frac{3}{2} \)
If \( t = \frac{3}{2} \):
\[ x - \frac{1}{x} = \frac{3}{2} \]
Multiplying through by \( x \) gives:
\[ x^2 - \frac{3}{2}x - 1 = 0 \]
To solve this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -\frac{3}{2}, c = -1 \):
\[ x = \frac{\frac{3}{2} \pm \sqrt{(-\frac{3}{2})^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-\frac{3}{2})^2 = \frac{9}{4}, \quad 4 \cdot 1 \cdot (-1) = -4 \implies -4 = -\frac{16}{4} \implies \frac{9}{4} + \frac{16}{4} = \frac{25}{4} \]
Thus:
\[ x = \frac{\frac{3}{2} \pm \frac{5}{2}}{2} \]
This gives us two solutions:
1. \( x = \frac{8/2}{2} = 2 \)
2. \( x = \frac{-2/2}{2} = -\frac{1}{2} \)

### Step 7: Compile all solutions
The solutions we found are:
- From \( t = 0 \): \( x = 1, -1 \)
- From \( t = \frac{3}{2} \): \( x = 2, -\frac{1}{2} \)

Thus, the complete solution set is:
\[ \{ 1, -1, 2, -\frac{1}{2} \} \]