## Solving the Quadratic Equation: (x² - 2)² - 10(x² - 2) + 21 = 0

This equation might look intimidating at first, but we can solve it by using a simple substitution technique.

### Step 1: Substitution

Let's substitute **y = x² - 2**. This will transform the equation into a more familiar quadratic form:

**y² - 10y + 21 = 0**

### Step 2: Solving the Quadratic Equation

Now, we have a standard quadratic equation in terms of 'y'. We can solve this using the quadratic formula:

**y = [-b ± √(b² - 4ac)] / 2a**

Where:

- a = 1
- b = -10
- c = 21

Plugging these values into the formula:

**y = [10 ± √((-10)² - 4 * 1 * 21)] / 2 * 1**

**y = [10 ± √(100 - 84)] / 2**

**y = [10 ± √16] / 2**

**y = [10 ± 4] / 2**

This gives us two possible solutions for 'y':

**y1 = 7****y2 = 3**

### Step 3: Back Substitution

Now, we need to substitute back 'x² - 2' for 'y' in both solutions:

**x² - 2 = 7****x² - 2 = 3**

Solving for 'x' in both equations:

**x² = 9****x = ±3**

**x² = 5****x = ±√5**

### Final Solutions

Therefore, the solutions for the equation **(x² - 2)² - 10(x² - 2) + 21 = 0** are:

**x = 3****x = -3****x = √5****x = -√5**