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

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.