Sure! Let's find the solutions to the equation [tex]\( 12x^3 - 24x^2 + 12x = 0 \)[/tex].
1. Factor out the greatest common factor (GCF):
The GCF of the terms [tex]\(12x^3\)[/tex], [tex]\( -24x^2\)[/tex], and [tex]\( 12x \)[/tex] is [tex]\(12x\)[/tex]. So we factor [tex]\(12x\)[/tex] out of the equation:
[tex]\[
12x(x^2 - 2x + 1) = 0
\][/tex]
2. Simplify the equation:
Now, we have:
[tex]\[
12x(x^2 - 2x + 1) = 0
\][/tex]
3. Break the equation into separate factors:
For the product to be zero, at least one of the factors must be zero. So we have two potential cases:
[tex]\[
12x = 0 \quad \text{or} \quad x^2 - 2x + 1 = 0
\][/tex]
4. Solve the first factor:
For [tex]\(12x = 0\)[/tex]:
[tex]\[
x = 0
\][/tex]
5. Solve the second factor:
We need to solve [tex]\(x^2 - 2x + 1 = 0\)[/tex]. This is a quadratic equation that can be factored as follows:
[tex]\[
(x - 1)^2 = 0
\][/tex]
Setting the factor equal to zero gives us:
[tex]\[
x - 1 = 0 \implies x = 1
\][/tex]
6. Combine the solutions:
The solutions to the equation [tex]\( 12x^3 - 24x^2 + 12x = 0 \)[/tex] are thus:
[tex]\[
x = 0 \quad \text{or} \quad x = 1
\][/tex]
Therefore, the solutions to the equation [tex]\( 12x^3 - 24x^2 + 12x = 0 \)[/tex] are [tex]\( \boxed{0 \text{ and } 1} \)[/tex].
