To factorize the quadratic expression \(3x^2 - 7x + 2\), follow these steps:

1. **Identify the coefficients**:
   - \(a = 3\), \(b = -7\), and \(c = 2\) in the standard quadratic form \(ax^2 + bx + c\).

2. **Multiply \(a\) and \(c\)**:
   Multiply the first and last coefficients: \(3 \times 2 = 6\).

3. **Find two numbers that multiply to 6 and add to \(-7\)**:
   The numbers are \(-6\) and \(-1\), because:
   \[
   -6 \times -1 = 6 \quad \text{and} \quad -6 + (-1) = -7.
   \]

4. **Rewrite the middle term using these two numbers**:
   Rewrite \(-7x\) as \(-6x - x\):
   \[
   3x^2 - 6x - x + 2.
   \]

5. **Factor by grouping**:
   Group the terms:
   \[
   (3x^2 - 6x) - (x - 2).
   \]
   Factor out the greatest common factor (GCF) from each group:
   \[
   3x(x - 2) - 1(x - 2).
   \]

6. **Factor out the common binomial**:
   Factor out the common binomial factor \((x - 2)\):
   \[
   (3x - 1)(x - 2).
   \]

Thus, the factorized form of \(3x^2 - 7x + 2\) is:
\[
(3x - 1)(x - 2).
\]

Would you like further explanation on any step?

Here are 5 related questions for deeper understanding:
1. How would you factor a quadratic expression where \(a = 1\), like \(x^2 - 5x + 6\)?
2. What happens when a quadratic cannot be factorized into integers?
3. How can you use the quadratic formula to solve the equation \(3x^2 - 7x + 2 = 0\)?
4. What is the significance of the discriminant in determining factorability?
5. Can you expand the factors \((3x - 1)(x - 2)\) to verify the solution?

**Tip**: Always check your factorization by expanding the factors back into the original expression to ensure correctness.

This page provides a detailed step-by-step guide on how to factor the quadratic expression 3x^2 - 7x + 2. Learn to use factoring by grouping and understand key algebraic concepts. Perfect for students in Grades 9-10.

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