To solve the equation $x^4 - 4x^2 + 4 = 0$, let us analyze and simplify it step by step:

Step 1: Substitution

Let $y = x^2$. This transforms the equation into: $y^2 - 4y + 4 = 0$

Step 2: Factoring

The quadratic equation can be factored as: $(y - 2)^2 = 0$

Step 3: Solve for $y$

From $(y - 2)^2 = 0$, we get: $y = 2$

Step 4: Back-substitution

Recall $y = x^2$. Substituting back, we have: $x^2 = 2$

Step 5: Solve for $x$

Solving for $x$, we find: $x = \pm\sqrt{2}$

Final Answer:

The solutions are: $x = \sqrt{2} \quad \text{and} \quad x = -\sqrt{2}$

Would you like further details or clarifications?

Here are five related questions for deeper understanding:

1. How can we verify the solutions for this polynomial equation?
2. What does the solution set represent graphically?
3. How would the solution differ if the equation had a cubic term?
4. Can this method be used to solve higher-degree polynomial equations?
5. What happens if the discriminant of the quadratic in $y$ was negative?

Tip: Always verify your solutions by substituting them back into the original equation to ensure correctness.