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Answer :
To solve the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex], let's follow a step-by-step approach:
1. Simplify the Equation:
First, we notice that the equation can be simplified by combining like terms. The terms [tex]\(6x^3\)[/tex] and [tex]\(90x^3\)[/tex] can be combined:
[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]
So the equation becomes:
[tex]\[
96x^3 - 48x^4 = 0
\][/tex]
2. Factor Out the Common Term:
Next, let's factor out the common factor from each term. Both terms have a factor of [tex]\(48x^3\)[/tex]:
[tex]\[
48x^3 (2 - x) = 0
\][/tex]
3. Solve the Factored Equation:
An equation that is factored and set to zero means each factor must equal zero. Therefore, we have two separate equations to solve:
[tex]\[
48x^3 = 0
\][/tex]
and
[tex]\[
2 - x = 0
\][/tex]
- Solving [tex]\(48x^3 = 0\)[/tex]:
Divide both sides by 48, which gives [tex]\(x^3 = 0\)[/tex]. Taking the cube root on both sides results in [tex]\(x = 0\)[/tex].
- Solving [tex]\(2 - x = 0\)[/tex]:
Adding [tex]\(x\)[/tex] to both sides gives [tex]\(2 = x\)[/tex], or equivalently [tex]\(x = 2\)[/tex].
4. Conclusion:
The solution to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] is the set of values [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
1. Simplify the Equation:
First, we notice that the equation can be simplified by combining like terms. The terms [tex]\(6x^3\)[/tex] and [tex]\(90x^3\)[/tex] can be combined:
[tex]\[
6x^3 + 90x^3 = 96x^3
\][/tex]
So the equation becomes:
[tex]\[
96x^3 - 48x^4 = 0
\][/tex]
2. Factor Out the Common Term:
Next, let's factor out the common factor from each term. Both terms have a factor of [tex]\(48x^3\)[/tex]:
[tex]\[
48x^3 (2 - x) = 0
\][/tex]
3. Solve the Factored Equation:
An equation that is factored and set to zero means each factor must equal zero. Therefore, we have two separate equations to solve:
[tex]\[
48x^3 = 0
\][/tex]
and
[tex]\[
2 - x = 0
\][/tex]
- Solving [tex]\(48x^3 = 0\)[/tex]:
Divide both sides by 48, which gives [tex]\(x^3 = 0\)[/tex]. Taking the cube root on both sides results in [tex]\(x = 0\)[/tex].
- Solving [tex]\(2 - x = 0\)[/tex]:
Adding [tex]\(x\)[/tex] to both sides gives [tex]\(2 = x\)[/tex], or equivalently [tex]\(x = 2\)[/tex].
4. Conclusion:
The solution to the equation [tex]\(6x^3 - 48x^4 + 90x^3 = 0\)[/tex] is the set of values [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Therefore, the solutions are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
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