Answer :

Sure! Let's solve the equation [tex]\(25x^3 = 64x\)[/tex] step by step.

1. Bring all terms to one side of the equation:
[tex]\[
25x^3 - 64x = 0
\][/tex]

2. Factor out the common term:
Notice that there is a common factor of [tex]\(x\)[/tex] in each term. So, factor out [tex]\(x\)[/tex]:
[tex]\[
x(25x^2 - 64) = 0
\][/tex]

3. Apply the zero-product property:
The zero-product property tells us that if a product of factors equals zero, then at least one of the factors must be zero. This gives us:
[tex]\[
x = 0 \quad \text{or} \quad 25x^2 - 64 = 0
\][/tex]

4. Solve the quadratic equation:
- Let's solve [tex]\(25x^2 - 64 = 0\)[/tex].

Add 64 to both sides:
[tex]\[
25x^2 = 64
\][/tex]

Divide both sides by 25:
[tex]\[
x^2 = \frac{64}{25}
\][/tex]

Take the square root of both sides:
[tex]\[
x = \pm\sqrt{\frac{64}{25}}
\][/tex]

Simplify the square root:
[tex]\[
x = \pm\frac{8}{5}
\][/tex]

5. Combine all solutions:
The solutions to the equation [tex]\(25x^3 = 64x\)[/tex] are:
[tex]\[
x = 0, \quad x = \frac{8}{5}, \quad x = -\frac{8}{5}
\][/tex]

These are all the solutions for the given equation.

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