## Solving the Equation (x-3)^4 - 5(x-3)^2 - 36 = 0

This equation may seem intimidating at first glance, but it can be solved efficiently using a simple substitution technique. Let's break down the steps:

### 1. Substitute and Simplify

We can simplify the equation by substituting a new variable. Let **y = (x-3)**. Substituting this into the equation, we get:

**y^4 - 5y^2 - 36 = 0**

This is now a quadratic equation in terms of 'y'.

### 2. Factor the Quadratic

The quadratic equation can be factored as follows:

**(y^2 - 9)(y^2 + 4) = 0**

This gives us two factors:

**y^2 - 9 = 0****y^2 + 4 = 0**

### 3. Solve for 'y'

Solving for 'y' in each factor:

**y^2 - 9 = 0**=>**y = ±3****y^2 + 4 = 0**=>**y = ±2i**(where 'i' is the imaginary unit, √-1)

### 4. Substitute back to find 'x'

Now we substitute back **y = (x-3)** to find the values of 'x':

**y = 3:**- (x-3) = 3
**x = 6**

**y = -3:**- (x-3) = -3
**x = 0**

**y = 2i:**- (x-3) = 2i
**x = 3 + 2i**

**y = -2i:**- (x-3) = -2i
**x = 3 - 2i**

### Conclusion

Therefore, the solutions to the equation (x-3)^4 - 5(x-3)^2 - 36 = 0 are:

**x = 6****x = 0****x = 3 + 2i****x = 3 - 2i**