Answer :

To solve the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex], we will follow these steps:

1. Combine Like Terms: Start by combining any like terms in the equation. In this case, we have the terms [tex]\(6x^3\)[/tex] and [tex]\(90x^3\)[/tex], which can be combined:

[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]

Now, the equation becomes:

[tex]\[
96x^3 - 48x^4 = 0
\][/tex]

2. Factor Out the Greatest Common Factor: Look for the greatest common factor (GCF) in the terms. Here, the GCF is [tex]\(48x^3\)[/tex]. Factor this out:

[tex]\[
48x^3(2 - x) = 0
\][/tex]

3. Apply the Zero Product Property: According to the zero product property, if the product of two expressions is zero, at least one of the expressions must be zero. Therefore, set each factor equal to zero:

[tex]\[
48x^3 = 0 \quad \text{or} \quad (2 - x) = 0
\][/tex]

4. Solve Each Equation:

- For [tex]\(48x^3 = 0\)[/tex]: Divide by 48 to obtain:

[tex]\[
x^3 = 0
\][/tex]

Taking the cube root of both sides, we find:

[tex]\[
x = 0
\][/tex]

- For [tex]\(2 - x = 0\)[/tex]: Solve for [tex]\(x\)[/tex] by adding [tex]\(x\)[/tex] to both sides and rearranging:

[tex]\[
x = 2
\][/tex]

5. Write the Solutions: Therefore, the solutions to the equation are:

[tex]\[
x = 0 \quad \text{and} \quad x = 2
\][/tex]

These are the values of [tex]\(x\)[/tex] that satisfy the original equation.