To solve the inequality \( x^4 - 2x^2 \geq 8 \), let's break it down step by step.

### Step 1: Rewrite the inequality
The original inequality is:

\[
x^4 - 2x^2 \geq 8
\]

Subtract 8 from both sides:

\[
x^4 - 2x^2 - 8 \geq 0
\]

### Step 2: Substitute
To make it simpler, let's make a substitution. Let \( y = x^2 \). Then, the inequality becomes:

\[
y^2 - 2y - 8 \geq 0
\]

Now, we need to solve this quadratic inequality in \( y \).

### Step 3: Solve the quadratic equation
First, solve the quadratic equation \( y^2 - 2y - 8 = 0 \) using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -2 \), and \( c =

Learn how to solve the inequality x^4 - 2x^2 ≥ 8 with a detailed step-by-step solution. This problem involves inequalities and quadratic equations, specifically addressing how to manipulate and solve such equations algebraically. Suitable for advanced high school students.

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