Solve x
in the inequality
x^4−4⋅x^2−45≥0

To solve the inequality \( x^4 - 4x^2 - 45 \geq 0 \), let's start by rewriting the expression in a more manageable form.

### Step 1: Substitution
Let \( y = x^2 \). Then the inequality becomes:
\[
y^2 - 4y - 45 \geq 0
\]
This is a quadratic inequality in terms of \( y \).

### Step 2: Solve the quadratic equation
To find the critical points, solve the equation:
\[
y^2 - 4y - 45 = 0
\]
Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -4 \), and \( c = -45 \):
\[
y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-45)}}{2(1)}
\]
\[
y = \frac{4 \pm \sqrt{16 + 180}}{2}
\]
\[
y = \frac{4 \pm \sqrt{196}}{2}
\]
\[
y = \frac{4 \pm 14}{2}
\]
\[
y = \frac{4 + 14}{2} \quad \text{or} \quad y = \frac{4 - 14}{2}
\]
\[
y = 9 \quad \text{or} \quad y = -5
\]

### Step 3: Consider the valid roots
Since \( y = x^2 \), \( y \) must be non-negative. Therefore, \( y = -5 \) is not valid, and we only consider \( y = 9 \).

So the quadratic inequality reduces to:
\[
(x^2 - 9)(x^2 + 5) \geq 0
\]

### Step 4: Analyze the factors
1. \( x^2 - 9 \geq 0 \) implies \( x^2 \geq 9 \), so \( x \leq -3 \) or \( x \geq 3 \).
2. \( x^2 + 5 \geq 0 \) is always true for all \( x \) since \( x^2 \) is always non-negative, and adding 5 keeps it positive.

### Step 5: Determine the solution set
The inequality holds when \( x^2 - 9 \geq 0 \), which gives us the intervals:
\[
x \in (-\infty, -3] \cup [3, \infty)
\]

### Final Answer
The solution to the inequality \( x^4 - 4x^2 - 45 \geq 0 \) is:
\[
x \in (-\infty, -3] \cup [3, \infty)
\]

Would you like further details or have any questions? Here are some related problems you might consider:

1. Solve \( x^4 - 4x^2 - 45 = 0 \).
2. Determine the solution set for \( x^2 - 9 \leq 0 \).
3. Analyze the graph of \( y = x^4 - 4x^2 - 45 \).
4. Solve \( x^4 - 4x^2 + 45 \geq 0 \).
5. Determine the roots of the polynomial \( y = x^4 - 4x^2 - 45 \).
6. Explore the inequality \( x^4 + 4x^2 + 45 \geq 0 \).
7. Solve the quadratic inequality \( y^2 - 4y - 45 > 0 \).
8. Explore the properties of the function \( f(x) = x^4 - 4x^2 - 45 \).

**Tip:** When solving inequalities involving polynomials, always consider the sign changes of the expression across the critical points.

Learn how to solve the inequality x^4 - 4x^2 - 45 ≥ 0 with detailed step-by-step solutions and explanations. This problem involves polynomial inequalities and critical point analysis. Suitable for advanced high school students and college-level mathematics.

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