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 using the grouping method, also known as the AC method. Here's the solution step-by-step:
Step 1: Multiply and (from )
Here, , , and .
Step 2: Find two numbers that multiply to and add to
The two numbers are 6 and 2 because:
Step 3: Rewrite the middle term () using these two numbers
Step 4: Group terms in pairs
Step 5: Factor out the greatest common factor (GCF) from each group
Step 6: Factor out the common binomial factor
Final Factorization
The factorization of is:
Let me know if you'd like more explanation or a walkthrough of other examples!
Related Questions:
- How is the AC method applied for negative coefficients?
- What if no two numbers satisfy the product-sum relationship? What happens then?
- How does factoring by grouping compare to other factoring techniques?
- Can this quadratic expression be solved for its roots after factoring?
- 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