Let's factorize the expression \( 3x^2 - 12x + 9 \).

### Step 1: Identify the greatest common factor (GCF)
First, look for the GCF of the coefficients \(3\), \(12\), and \(9\). The GCF is \(3\).

### Step 2: Factor out the GCF
Factor out the \(3\) from the expression:
\[
3(x^2 - 4x + 3)
\]

### Step 3: Factor the quadratic expression
Now, we need to factor the quadratic expression \(x^2 - 4x + 3\).

We need two numbers that multiply to \(3\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). These numbers are \(-1\) and \(-3\).

So, the quadratic expression can be written as:
\[
x^2 - 4x + 3 = (x - 1)(x - 3)
\]

### Step 4: Write the final factorized form
Substitute back into the expression:
\[
3(x - 1)(x - 3)
\]

So, the fully factorized form of \( 3x^2 - 12x + 9 \) is:
\[
\boxed{3(x - 1)(x - 3)}
\]

Would you like more details or have any questions?

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Here are some related questions you might find interesting:
1. How would you factorize a quadratic trinomial like \(x^2 - 4x + 3\) if the GCF is 1?
2. What are the roots of the quadratic equation \(3x^2 - 12x + 9 = 0\)?
3. How can you check if your factorization of a quadratic expression is correct?
4. What does it mean for a quadratic equation to have repeated roots?
5. How would you apply the quadratic formula to solve \(3x^2 - 12x + 9 = 0\)?
6. How would you factor a quadratic expression with no real roots?
7. What is the significance of the discriminant in determining the nature of the roots?
8. Can you use factoring to solve quadratic inequalities?

**Tip:** After factorizing a quadratic expression, you can expand the factors to check if you get the original expression back, ensuring that the factorization is correct.

Learn how to factorize the quadratic expression 3x^2 - 12x + 9 using the greatest common factor and factoring techniques. This problem covers key algebraic concepts, including factoring quadratic expressions and identifying common factors. Suitable for students in Grades 9-12.

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