#### Answer

$\left\{ -\dfrac{8}{5},0,\dfrac{8}{5} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
Express the given equation, $
25x^3=64x
,$ in factored form. Then use the Zero Product Property by equating each factor to zero. Finally, solve each of the resulting equations.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
25x^3-64x=0
.\end{array}
The $GCF$ of the constants of the terms $\{
25,-64
\}$ is $
1
$ since it is the highest number that can divide all the given constants. The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
x^3,x
\}$ is $
x
.$ Hence, the entire expression has $GCF=
x
.$
Factoring the $GCF=
x
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(25x^2-64)=0
.\end{array}
The expressions $
25x^2
$ and $
64
$ are both perfect squares and are separated by a minus sign. Hence, $
25x^2-64
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(5x+8)(5x-8)=0
.\end{array}
Equating each factor to zero (Zero Product Property), then
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
5x+8=0
\\\\\text{OR}\\\\
5x-8=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
5x+8=0
\\\\
5x=-8
\\\\
x=-\dfrac{8}{5}
\\\\\text{OR}\\\\
5x=8
\\\\
x=\dfrac{8}{5}
.\end{array}
Hence, the solutions are $
\left\{ -\dfrac{8}{5},0,\dfrac{8}{5} \right\}
.$