Answer :

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

1. Combine Like Terms:
Start by combining the like terms in the equation. Notice the terms with [tex]\(x^3\)[/tex]:
[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]

Now, the equation simplifies to:
[tex]\[
96x^3 - 48x^4 = 0
\][/tex]

2. Factor the Equation:
Look for common factors in the equation. Both terms have a common factor of [tex]\(48x^3\)[/tex]. Factor out [tex]\(48x^3\)[/tex]:
[tex]\[
48x^3(x - 2) = 0
\][/tex]

3. Apply the Zero Product Property:
According to the zero product property, if a product of factors is equal to zero, at least one of the factors must be zero. So, set each factor to zero and solve:
[tex]\[
48x^3 = 0 \quad \text{or} \quad x - 2 = 0
\][/tex]

4. Solve Each Factor Separately:

- For [tex]\(48x^3 = 0\)[/tex]:
Divide both sides by 48 to get:
[tex]\[
x^3 = 0
\][/tex]
The only value of [tex]\(x\)[/tex] that satisfies [tex]\(x^3 = 0\)[/tex] is [tex]\(x = 0\)[/tex].

- For [tex]\(x - 2 = 0\)[/tex]:
Add 2 to both sides to find:
[tex]\[
x = 2
\][/tex]

5. Solutions:
The solutions to the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].

Therefore, the solution set for the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] is [tex]\([0, 2]\)[/tex].

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