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Solve the equation:

[tex]6x^3 - 48x^4 + 90x^3 = 0[/tex]

Answer :

Let's solve the equation step-by-step:

Given equation:
[tex]\[ 6x^3 - 48x^4 + 90x^3 = 0 \][/tex]

Step 1: Combine like terms.
Combine the terms with [tex]\( x^3 \)[/tex]:
[tex]\[ 6x^3 + 90x^3 = 96x^3 \][/tex]

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

Step 2: Factor the equation.
First, factor out the greatest common factor which is [tex]\( x^3 \)[/tex]:
[tex]\[ x^3(96 - 48x) = 0 \][/tex]

Step 3: Set each factor equal to zero and solve.

1. For [tex]\( x^3 = 0 \)[/tex]:
The solution is:
[tex]\[ x = 0 \][/tex]

2. For [tex]\( 96 - 48x = 0 \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
48x = 96
\][/tex]
Divide both sides by 48:
[tex]\[
x = \frac{96}{48} = 2
\][/tex]

Solution:
The solutions to the equation are:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = 2 \][/tex]

So, the values of [tex]\( x \)[/tex] that satisfy the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex].

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