## Solving the Equation: (x + 1/x)² - 3/2(x - 1/x) = 4

This article will guide you through solving the equation **(x + 1/x)² - 3/2(x - 1/x) = 4**. We will utilize algebraic manipulation and substitution to find the solutions for x.

### Step 1: Simplifying the Equation

Let's start by simplifying the equation. We can expand the square term and distribute the constant:

(x + 1/x)² - 3/2(x - 1/x) = 4

x² + 2 + 1/x² - 3/2x + 3/2x - 3/2 = 4

**x² + 1/x² - 1/2 = 4**

### Step 2: Introducing a Substitution

To make the equation easier to solve, let's introduce a substitution. Let **y = x + 1/x**. Squaring both sides of this equation, we get:

y² = (x + 1/x)² = x² + 2 + 1/x²

Notice that the expression **x² + 1/x²** appears in our original equation. We can substitute y² - 2 for this expression:

y² - 2 - 1/2 = 4

### Step 3: Solving for y

Now we have a quadratic equation in terms of y:

y² - 5/2 = 0

Solving for y, we get:

y² = 5/2

y = ±√(5/2)

### Step 4: Solving for x

We have two possible values for y. Let's substitute back to find the corresponding values of x:

**Case 1: y = √(5/2)**

x + 1/x = √(5/2)

Multiplying both sides by x, we get:

x² + 1 = √(5/2)x

Rearranging into a quadratic equation:

x² - √(5/2)x + 1 = 0

Solving this quadratic equation (using the quadratic formula or factoring), we get two solutions for x.

**Case 2: y = -√(5/2)**

Following the same procedure as in Case 1, we will obtain two more solutions for x.

### Conclusion

By simplifying the equation, using substitution, and solving for y and x, we can find the solutions to the equation (x + 1/x)² - 3/2(x - 1/x) = 4. Remember to check your solutions by plugging them back into the original equation to ensure they are valid.