Question

Use algebraic​ and/or graphical methods to solve the inequality.
25 x squared minus x Superscript 4 Baseline greater than or equals 0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
C.
There is no solution.

Answer 1

To solve the inequality \(25x^2 - x^4 \geq 0\), we can start by rewriting it:

\[
-x^4 + 25x^2 \geq 0
\]

This can be rearranged to:

\[
x^4 - 25x^2 \leq 0
\]

Next, we can factor the left-hand side:

\[
x^2(x^2 - 25) \leq 0
\]

We can factor further:

\[
x^2(x - 5)(x + 5) \leq 0
\]

Now we can analyze the expression \(x^2(x - 5)(x + 5)\). The zeros of the expression (the points where it equals zero) are \(x = 0\), \(x = 5\), and \(x = -5\).

These points split the number line into intervals. We need to test these intervals to determine where the expression is non-positive:

1. **Interval: \( (-\infty, -5) \)**
- Choose \( x = -6 \):
\[
(-6)^2(-6 - 5)(-6 + 5) = 36 \cdot (-11) \cdot (-1) = 396 > 0
\]
(Not included)

2. **Interval: \( (-5, 0) \)**
- Choose \( x = -1 \):
\[
(-1)^2(-1 - 5)(-1 + 5) = 1 \cdot (-6) \cdot 4 = -24 < 0
\]
(Included)

3. **Interval: \( (0, 5) \)**
- Choose \( x = 1 \):
\[
(1)^2(1 - 5)(1 + 5) = 1 \cdot (-4) \cdot 6 = -24 < 0
\]
(Included)

4. **Interval: \( (5, \infty) \)**
- Choose \( x = 6 \):
\[
(6)^2(6 - 5)(6 + 5) = 36 \cdot 1 \cdot 11 = 396 > 0
\]
(Not included)

Now, we include the points where the expression is equal to zero, which are \(x = -5\), \(x = 0\), and \(x = 5\).

Thus, the solution to the inequality \( 25x^2 - x^4 \geq 0 \) is:

\[
[-5, 0] \cup [0, 5]
\]

### Final Answer

**B. The solutions are the values of \(x\) in the interval:**
\[
[-5, 0] \cup [0, 5]
\]