Describe the error in the problem $$ x^{3}-25 x=0 $$ $$ x^{3}=25 x $$ $$ x^{2}=25 $$ $$ x= \pm 5 $$

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Let's analyze the steps provided in the problem to identify any errors.

1. The original equation is:
   $$
   x^{3} - 25x = 0
   $$
   This is correct.

2. The next step is:
   $$
   x^{3} = 25x
   $$
   This is also correct, as it is simply rearranging the original equation.

3. The next step is:
   $$
   x^{2} = 25
   $$
   This step is where the error occurs. To go from $$ x^{3} = 25x $$ to $$ x^{2} = 25 $$, we need to factor out $$ x $$ from the left side of the equation. The correct factorization should be:
   $$
   x(x^{2} - 25) = 0
   $$
   This gives us two cases to consider:
   - $$ x = 0 $$
   - $$ x^{2} - 25 = 0 $$

4. Solving $$ x^{2} - 25 = 0 $$ gives:
   $$
   x^{2} = 25 \implies x = \pm 5
   $$

Thus, the complete solution set for the equation $$ x^{3} - 25x = 0 $$ is:
$$
x = 0, \quad x = 5, \quad x = -5
$$

In summary, the error in the problem is the omission of the $$ x = 0 $$ solution and the incorrect transition from $$ x^{3} = 25x $$ to $$ x^{2} = 25 $$ without factoring. The correct approach should include factoring and considering all possible solutions.
The error is in the step where $$ x^{3} = 25x $$ is incorrectly simplified to $$ x^{2} = 25 $$. The correct approach is to factor out $$ x $$, resulting in $$ x(x^{2} - 25) = 0 $$, which gives solutions $$ x = 0 $$, $$ x = 5 $$, and $$ x = -5 $$.

Describe the error in the problem \( x^{3}-25 x=0 \) \( x^{3}=25 x \) \( x^{2}=25 \) \( x= \pm 5 \)

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