Factor the expression.
6x^(3⁄2) − 18x^(1⁄2) + 12x^(−1⁄2)

To factor the expression \(6x^{3/2} - 18x^{1/2} + 12x^{-1/2}\), we can start by finding the greatest common factor (GCF) of all the terms.

### Step 1: Identify the GCF

The terms are:
- \(6x^{3/2}\)
- \(-18x^{1/2}\)
- \(12x^{-1/2}\)

First, let's look at the numerical coefficients: \(6, -18, 12\). The GCF of these numbers is \(6\).

Next, we look at the variable \(x\) and its exponents: \( \frac{3}{2}, \frac{1}{2}, -\frac{1}{2} \). The GCF of the exponents is the smallest exponent, which is \(-\frac{1}{2}\).

Thus, the GCF of all terms is \(6x^{-1/2}\).

### Step 2: Factor out the GCF

Now, we factor out \(6x^{-1/2}\) from the expression:

\[
6x^{3/2} - 18x^{1/2} + 12x^{-1/2} = 6x^{-1/2} \left( x^2 - 3x + 2 \right)
\]

### Step 3: Factor the quadratic expression

Now, we factor the quadratic expression inside the parentheses:

\[
x^2 - 3x + 2
\]

To factor this, we look for two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). These numbers are \(-1\) and \(-2\).

So, we can write:

\[
x^2 - 3x + 2 = (x - 1)(x - 2)
\]

### Step 4: Write the fully factored form

Now, substitute this back into the expression:

\[
6x^{3/2} - 18x^{1/2} + 12x^{-1/2} = 6x^{-1/2} (x - 1)(x - 2)
\]

### Final Answer

The fully factored form of the expression is:

\[
\boxed{6x^{-1/2} (x - 1)(x - 2)}
\]

Would you like a detailed explanation of any step?

Here are 5 related questions you might find interesting:
1. How do you determine the greatest common factor (GCF) of algebraic terms?
2. How can you factor expressions with fractional exponents?
3. What is the process for factoring quadratic expressions?
4. How do negative exponents affect factoring?
5. How do you simplify expressions with fractional and negative exponents?

**Tip:** Always check your factored expression by multiplying it back out to ensure it matches the original!

Learn how to factor the expression 6x^(3/2) - 18x^(1/2) + 12x^(-1/2) with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring with fractional exponents and quadratic equations. Suitable for students in Grades 10-12.

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