## Solving the Equation: (x² + 1/x²) - 5(x + 1/x) + 6 = 0

This equation might seem intimidating at first glance, but we can solve it by using a clever substitution and factoring. Here's how:

### 1. Substitution:

Let's make a substitution to simplify the equation. We can define:

**y = x + 1/x**

Now, let's square both sides of this equation:

**y² = (x + 1/x)² = x² + 2(x)(1/x) + 1/x² = x² + 2 + 1/x²**

Notice that we can rearrange this to get:

**x² + 1/x² = y² - 2**

### 2. Rewriting the Equation:

Substituting these expressions back into our original equation, we get:

**(y² - 2) - 5y + 6 = 0**

This simplifies to a quadratic equation:

**y² - 5y + 4 = 0**

### 3. Solving the Quadratic:

Now, we can easily factor this quadratic:

**(y - 4)(y - 1) = 0**

This gives us two possible solutions for y:

**y = 4****y = 1**

### 4. Finding x:

Now, we need to find the values of x that correspond to these values of y. Let's remember our substitution:

**y = x + 1/x**

For **y = 4**:

**4 = x + 1/x**

Multiplying both sides by x:

**4x = x² + 1**

Rearranging into a quadratic:

**x² - 4x + 1 = 0**

Using the quadratic formula, we get:

**x = (4 ± √12)/2 = 2 ± √3**

For **y = 1**:

**1 = x + 1/x**

Following a similar procedure as above, we get:

**x² - x + 1 = 0**

Again, using the quadratic formula, we get:

**x = (1 ± √-3)/2 = (1 ± i√3)/2**

where 'i' represents the imaginary unit (√-1).

### Conclusion:

Therefore, the solutions to the equation (x² + 1/x²) - 5(x + 1/x) + 6 = 0 are:

**x = 2 + √3****x = 2 - √3****x = (1 + i√3)/2****x = (1 - i√3)/2**