To solve the equation [tex]\(3(x-4)(2x-3) = 0\)[/tex], follow these steps:
1. Understand the Equation:
The equation is a polynomial set to zero, [tex]\(3(x-4)(2x-3) = 0\)[/tex]. This can be solved using the zero-product property, which states that if a product of several factors is zero, at least one of the factors must be zero.
2. Set Each Factor to Zero:
We will set each factor inside the parentheses equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[x - 4 = 0\][/tex]
[tex]\[2x - 3 = 0\][/tex]
3. Solve Each Factor Separately:
- For the first factor, [tex]\(x - 4 = 0\)[/tex]:
[tex]\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\][/tex]
- For the second factor, [tex]\(2x - 3 = 0\)[/tex]:
[tex]\[
2x - 3 = 0 \quad \Rightarrow \quad 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}
\][/tex]
4. Identify the Solutions:
The values of [tex]\(x\)[/tex] that make the equation true are [tex]\(x = 4\)[/tex] and [tex]\(x = \frac{3}{2}\)[/tex].
5. Checking Solutions Against Provided Options:
- [tex]\(-4\)[/tex]: This is not a solution.
- [tex]\(-3\)[/tex]: This is not a solution.
- [tex]\(-\frac{2}{3}\)[/tex]: This is not a solution.
- [tex]\(\frac{3}{2}\)[/tex]: This is a solution.
- 3: This is not a solution.
- 4: This is a solution.
Therefore, the solutions to the equation [tex]\(3(x-4)(2x-3) = 0\)[/tex] are [tex]\(\frac{3}{2}\)[/tex] and 4. The correct options are:
[tex]\[
\boxed{\frac{3}{2}} \quad \text{and} \quad \boxed{4}
\][/tex]
