Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Factoring Quadratic Expressions
**Problem 5: Factor the expression**
\[ 4x^2 - 39x + 27 \]
This task involves factoring a quadratic expression of the form \( ax^2 + bx + c \). The goal is to express the quadratic as a product of two binomials. Here, \( a = 4 \), \( b = -39 \), and \( c = 27 \).
**Steps to Factor the Quadratic:**
1. **Identify Coefficients**:
- The coefficient of \( x^2 \) is \( a = 4 \).
- The coefficient of \( x \) is \( b = -39 \).
- The constant term is \( c = 27 \).
2. **Multiply \( a \) and \( c \)**:
\[
ac = 4 \times 27 = 108
\]
3. **Find Two Numbers That Multiply to \( ac \) and Add to \( b \)**:
- We need two numbers whose product is \( 108 \) and whose sum is \( -39 \).
4. **Split the Middle Term**:
- Once the two numbers are found, \( b \) can be split accordingly.
5. **Factor by Grouping**:
- Group terms and factor common factors.
6. **Verify the Factored Form**:
- Expand to check correctness.
This approach can solve similar quadratic expressions effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa02accf-e7e0-450c-b255-e64057d9e798%2Fa7bd0d44-e938-449b-beee-de88ccef4682%2Fmbryc1.jpeg&w=3840&q=75)
Transcribed Image Text:### Factoring Quadratic Expressions
**Problem 5: Factor the expression**
\[ 4x^2 - 39x + 27 \]
This task involves factoring a quadratic expression of the form \( ax^2 + bx + c \). The goal is to express the quadratic as a product of two binomials. Here, \( a = 4 \), \( b = -39 \), and \( c = 27 \).
**Steps to Factor the Quadratic:**
1. **Identify Coefficients**:
- The coefficient of \( x^2 \) is \( a = 4 \).
- The coefficient of \( x \) is \( b = -39 \).
- The constant term is \( c = 27 \).
2. **Multiply \( a \) and \( c \)**:
\[
ac = 4 \times 27 = 108
\]
3. **Find Two Numbers That Multiply to \( ac \) and Add to \( b \)**:
- We need two numbers whose product is \( 108 \) and whose sum is \( -39 \).
4. **Split the Middle Term**:
- Once the two numbers are found, \( b \) can be split accordingly.
5. **Factor by Grouping**:
- Group terms and factor common factors.
6. **Verify the Factored Form**:
- Expand to check correctness.
This approach can solve similar quadratic expressions effectively.
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