To find the values of [tex]\(x\)[/tex] in the equation [tex]\(3x^3 = 12x\)[/tex], follow these steps:
1. Rewrite the equation:
[tex]\[ 3x^3 = 12x \][/tex]
2. Move all terms to one side to set the equation to zero:
[tex]\[ 3x^3 - 12x = 0 \][/tex]
3. Factor out the common term:
Notice that both terms on the left side of the equation have a common factor of [tex]\(3x\)[/tex]:
[tex]\[ 3x(x^2 - 4) = 0 \][/tex]
4. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
We have two factors to consider:
[tex]\[ 3x = 0 \][/tex]
[tex]\[ x^2 - 4 = 0 \][/tex]
- For the first factor [tex]\(3x = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
- For the second factor [tex]\( x^2 - 4 = 0 \)[/tex]:
Rewrite the equation to find [tex]\(x\)[/tex]:
[tex]\[ x^2 = 4 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm 2 \][/tex]
This gives two solutions:
[tex]\[ x = 2 \][/tex]
[tex]\[ x = -2 \][/tex]
5. List all solutions:
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(3x^3 = 12x\)[/tex] are:
[tex]\[ 0, 2, \text{ and } -2 \][/tex]
Thus, the correct answer is [tex]\(b. \, 2, -2, 0\)[/tex].
