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 any like terms in the expression. In this equation, [tex]\(6x^3\)[/tex] and [tex]\(90x^3\)[/tex] are like terms, so add them together:
[tex]\[
(6x^3 + 90x^3) - 48x^4 = 96x^3 - 48x^4
\][/tex]
2. Factor the Equation:
Next, factor out the greatest common factor from the terms in the equation. The greatest common factor here is [tex]\( 48x^3 \)[/tex] because it is divisible by both terms:
[tex]\[
x^3(96 - 48x) = 0
\][/tex]
3. Set Each Factor to Zero:
For the product of factors to be zero, at least one of the factors must be zero. So, set each factor equal to zero:
[tex]\[
x^3 = 0 \quad \text{or} \quad 96 - 48x = 0
\][/tex]
The first factor, [tex]\(x^3 = 0\)[/tex], results in:
[tex]\[
x = 0
\][/tex]
For the second factor, solve for [tex]\(x\)[/tex]:
[tex]\[
96 - 48x = 0
\][/tex]
[tex]\[
48x = 96
\][/tex]
[tex]\[
x = \frac{96}{48} = 2
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the values of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
1. Combine Like Terms:
Start by combining any like terms in the expression. In this equation, [tex]\(6x^3\)[/tex] and [tex]\(90x^3\)[/tex] are like terms, so add them together:
[tex]\[
(6x^3 + 90x^3) - 48x^4 = 96x^3 - 48x^4
\][/tex]
2. Factor the Equation:
Next, factor out the greatest common factor from the terms in the equation. The greatest common factor here is [tex]\( 48x^3 \)[/tex] because it is divisible by both terms:
[tex]\[
x^3(96 - 48x) = 0
\][/tex]
3. Set Each Factor to Zero:
For the product of factors to be zero, at least one of the factors must be zero. So, set each factor equal to zero:
[tex]\[
x^3 = 0 \quad \text{or} \quad 96 - 48x = 0
\][/tex]
The first factor, [tex]\(x^3 = 0\)[/tex], results in:
[tex]\[
x = 0
\][/tex]
For the second factor, solve for [tex]\(x\)[/tex]:
[tex]\[
96 - 48x = 0
\][/tex]
[tex]\[
48x = 96
\][/tex]
[tex]\[
x = \frac{96}{48} = 2
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the values of [tex]\(x\)[/tex] that satisfy the equation are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].