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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.
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.
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