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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Question
![**Factoring Quadratic Expressions**
The problem presented is to factor the quadratic expression:
\[ 3x^2 + 8x + 5 \]
The formatted expression shows how it can be rewritten as the product of two binomials:
\[ ( \_\_ x + \_\_ )( \_\_ x + \_\_ ) \]
To solve this, follow these steps to find the correct values for the blanks:
1. **Identify the Coefficients**:
The expression is in the form of \( ax^2 + bx + c \), where \( a = 3 \), \( b = 8 \), and \( c = 5 \).
2. **Factorization Structure**:
We need to find two numbers whose product is \( a \times c = 3 \times 5 = 15 \) and whose sum is \( b = 8 \).
3. **Find the Numbers**:
The numbers 3 and 5 fit because \( 3 \times 5 = 15 \) and \( 3 + 5 = 8 \).
4. **Rewrite and Factor**:
Using these numbers, rewrite and group the expression:
\[ 3x^2 + 3x + 5x + 5 \]
Factor by grouping:
\[ 3x(x + 1) + 5(x + 1) \]
Finally, factor out the common binomial:
\[ (3x + 5)(x + 1) \]
5. **Verification**:
Expand to check:
\[ (3x + 5)(x + 1) = 3x^2 + 3x + 5x + 5 = 3x^2 + 8x + 5 \]
The factored form is \( (3x + 5)(x + 1) \).
Click the "Submit Question" button to validate your solution.
This approach helps in understanding how to break down and solve quadratic expressions effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f3023e4-e765-481d-a4cf-2667beaa33ac%2Fcabe04ca-6327-4ed4-9bc2-4398133710af%2Feod9j1a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Factoring Quadratic Expressions**
The problem presented is to factor the quadratic expression:
\[ 3x^2 + 8x + 5 \]
The formatted expression shows how it can be rewritten as the product of two binomials:
\[ ( \_\_ x + \_\_ )( \_\_ x + \_\_ ) \]
To solve this, follow these steps to find the correct values for the blanks:
1. **Identify the Coefficients**:
The expression is in the form of \( ax^2 + bx + c \), where \( a = 3 \), \( b = 8 \), and \( c = 5 \).
2. **Factorization Structure**:
We need to find two numbers whose product is \( a \times c = 3 \times 5 = 15 \) and whose sum is \( b = 8 \).
3. **Find the Numbers**:
The numbers 3 and 5 fit because \( 3 \times 5 = 15 \) and \( 3 + 5 = 8 \).
4. **Rewrite and Factor**:
Using these numbers, rewrite and group the expression:
\[ 3x^2 + 3x + 5x + 5 \]
Factor by grouping:
\[ 3x(x + 1) + 5(x + 1) \]
Finally, factor out the common binomial:
\[ (3x + 5)(x + 1) \]
5. **Verification**:
Expand to check:
\[ (3x + 5)(x + 1) = 3x^2 + 3x + 5x + 5 = 3x^2 + 8x + 5 \]
The factored form is \( (3x + 5)(x + 1) \).
Click the "Submit Question" button to validate your solution.
This approach helps in understanding how to break down and solve quadratic expressions effectively.
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