To find \( x \) in the equation \( x^4 - 5x^2 + 4 = 0 \), we can treat it as a quadratic equation in terms of \( x^2 \). Let’s solve it step by step:

### Step 1: Substitution
Let \( y = x^2 \). This transforms the equation into:

\[
y^2 - 5y + 4 = 0
\]

### Step 2: Solve the quadratic equation
We solve \( y^2 - 5y + 4 = 0 \) using the quadratic formula:

\[
y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)}
\]

\[
y = \frac{5 \pm \sqrt{25 - 16}}{2}
\]

\[
y = \frac{5 \pm \sqrt{9}}{2}
\]

\[
y = \frac{5 \pm 3}{2}
\]

So, we get two solutions for \( y \):

\[
y_1 = \frac{5 + 3}{2} = 4 \quad \text{and} \quad y_2 = \frac{5 - 3}{2} = 1
\]

### Step 3: Return to \( x \)
Now that we have \( y_1 = 4 \) and \( y_2 = 1 \), we substitute back \( y = x^2 \):

\[
x^2 = 4 \quad \Rightarrow \quad x = \pm 2
\]
\[
x^2 = 1 \quad \Rightarrow \quad x = \pm 1
\]

### Final Solution:
The solutions for \( x \) are:
\[
x = \pm 2, \pm 1
\]

So, \( x = 2, -2, 1, -1 \).

Would you like a detailed explanation of any step or have further questions?

Here are 5 related questions you could explore:
1. How would you solve \( x^4 - 7x^2 + 6 = 0 \)?
2. What is the quadratic formula, and how is it derived?
3. Can you factor the equation \( x^4 -

Learn how to solve the polynomial equation x^4 - 5x^2 + 4 by substituting y = x^2 and using the quadratic formula. This step-by-step guide covers core algebraic concepts, such as solving quadratic-like polynomial equations. Suitable for high school students in Grades 9-12.

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