Math Problem Statement

Factor the quadratic expression 4x^2 + 8x + 3 using the grouping method (AC method).

Solution

The problem requires factoring the quadratic expression 4x2+8x+34x^2 + 8x + 3 using the grouping method, also known as the AC method. Here's the solution step-by-step:

Step 1: Multiply aa and cc (from ax2+bx+cax^2 + bx + c)

Here, a=4a = 4, b=8b = 8, and c=3c = 3. ac=43=12a \cdot c = 4 \cdot 3 = 12

Step 2: Find two numbers that multiply to ac=12a \cdot c = 12 and add to b=8b = 8

The two numbers are 6 and 2 because: 62=12and6+2=86 \cdot 2 = 12 \quad \text{and} \quad 6 + 2 = 8

Step 3: Rewrite the middle term (8x8x) using these two numbers

4x2+6x+2x+34x^2 + 6x + 2x + 3

Step 4: Group terms in pairs

(4x2+6x)+(2x+3)(4x^2 + 6x) + (2x + 3)

Step 5: Factor out the greatest common factor (GCF) from each group

2x(2x+3)+1(2x+3)2x(2x + 3) + 1(2x + 3)

Step 6: Factor out the common binomial factor

(2x+3)(2x+1)(2x + 3)(2x + 1)

Final Factorization

The factorization of 4x2+8x+34x^2 + 8x + 3 is: (2x+3)(2x+1)(2x + 3)(2x + 1)

Let me know if you'd like more explanation or a walkthrough of other examples!


Related Questions:

  1. How is the AC method applied for negative coefficients?
  2. What if no two numbers satisfy the product-sum relationship? What happens then?
  3. How does factoring by grouping compare to other factoring techniques?
  4. Can this quadratic expression be solved for its roots after factoring?
  5. How do we verify if our factorization is correct?

Tip:

Always double-check your factorization by expanding the factors to ensure you get back to the original expression.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring Quadratic Expressions
AC Method

Formulas

Standard form of a quadratic equation: ax^2 + bx + c
Factoring by grouping: Splitting the middle term

Theorems

Distributive property of multiplication over addition
Zero-product property (for solving equations)

Suitable Grade Level

Grades 8-10