Answer :
Let's solve the equation [tex]\(96x^3 - 48x^4 = 0\)[/tex] step by step.
1. Factor the equation:
The first step is to factor out the greatest common factor (GCF) from the expression. We notice that both terms, [tex]\(96x^3\)[/tex] and [tex]\(48x^4\)[/tex], have a common factor. The GCF is [tex]\(48x^3\)[/tex].
So, we can write:
[tex]\[
96x^3 - 48x^4 = 48x^3(2 - x)
\][/tex]
2. Set each factor equal to zero:
Now that we have factored the equation into two parts, we can set each factor equal to zero:
a. [tex]\(48x^3 = 0\)[/tex]
b. [tex]\(2 - x = 0\)[/tex]
3. Solve each equation separately:
a. Solve [tex]\(48x^3 = 0\)[/tex]:
Divide both sides by [tex]\(48\)[/tex] to isolate [tex]\(x^3\)[/tex]:
[tex]\[
x^3 = 0
\][/tex]
To find [tex]\(x\)[/tex], take the cube root of both sides:
[tex]\[
x = 0
\][/tex]
So, one solution is [tex]\(x = 0\)[/tex].
b. Solve [tex]\(2 - x = 0\)[/tex]:
To isolate [tex]\(x\)[/tex], add [tex]\(x\)[/tex] to both sides:
[tex]\[
2 = x
\][/tex]
This equation simplifies to:
[tex]\[
x = 2
\][/tex]
Thus, another solution is [tex]\(x = 2\)[/tex].
4. Conclusion:
The solutions to the equation [tex]\(96x^3 - 48x^4 = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex]. These are the values of [tex]\(x\)[/tex] that satisfy the original equation.
1. Factor the equation:
The first step is to factor out the greatest common factor (GCF) from the expression. We notice that both terms, [tex]\(96x^3\)[/tex] and [tex]\(48x^4\)[/tex], have a common factor. The GCF is [tex]\(48x^3\)[/tex].
So, we can write:
[tex]\[
96x^3 - 48x^4 = 48x^3(2 - x)
\][/tex]
2. Set each factor equal to zero:
Now that we have factored the equation into two parts, we can set each factor equal to zero:
a. [tex]\(48x^3 = 0\)[/tex]
b. [tex]\(2 - x = 0\)[/tex]
3. Solve each equation separately:
a. Solve [tex]\(48x^3 = 0\)[/tex]:
Divide both sides by [tex]\(48\)[/tex] to isolate [tex]\(x^3\)[/tex]:
[tex]\[
x^3 = 0
\][/tex]
To find [tex]\(x\)[/tex], take the cube root of both sides:
[tex]\[
x = 0
\][/tex]
So, one solution is [tex]\(x = 0\)[/tex].
b. Solve [tex]\(2 - x = 0\)[/tex]:
To isolate [tex]\(x\)[/tex], add [tex]\(x\)[/tex] to both sides:
[tex]\[
2 = x
\][/tex]
This equation simplifies to:
[tex]\[
x = 2
\][/tex]
Thus, another solution is [tex]\(x = 2\)[/tex].
4. Conclusion:
The solutions to the equation [tex]\(96x^3 - 48x^4 = 0\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex]. These are the values of [tex]\(x\)[/tex] that satisfy the original equation.
