Solve the equation. Express numbers in exact simplest form. $$ x^{3}-49 x=0 $$ The solution set is $$ \{\square\} $$.

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UpStudy AI · Accepted Answer

Factor the given equation:
$$
x^{3} - 49x = 0.
$$

Step 1: Factor out $$ x $$:
$$
x(x^{2} - 49) = 0.
$$

Step 2: Recognize $$ x^{2} - 49 $$ as a difference of two squares:
$$
x^{2} - 49 = (x-7)(x+7).
$$

Step 3: Write the fully factored form of the equation:
$$
x(x-7)(x+7)=0.
$$

Step 4: Set each factor equal to zero and solve for $$ x $$:
$$
\begin{cases}
x = 0, \
x - 7 = 0 \quad \Rightarrow \quad x = 7, \
x + 7 = 0 \quad \Rightarrow \quad x = -7.
\end{cases}
$$

Thus, the solution set is 
$$
\{-7, 0, 7\}.
$$
The solutions are $$ x = -7 $$, $$ x = 0 $$, and $$ x = 7 $$.

Solve the equation. Express numbers in exact simplest form. \[ x^{3}-49 x=0 \] The solution set is \( \{\square\} \).

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