Use algebraic​ and/or graphical methods to solve the inequality.

Question

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