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Factor the following quadratic: 3x²+ 13x + 4

Factor the following quadratic: 3x²+ 13x + 4

Algebra and Trigonometry (6th Edition)
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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**Quadratic Factorization Exercise** --- **Problem Statement:** Factor the following quadratic expression: \( 3x^2 + 13x + 4 \) **Instructions:** To factor this quadratic expression, follow these steps: 1. **Identify the coefficients** in the quadratic equation of the form \( ax^2 + bx + c \): - \( a = 3 \) - \( b = 13 \) - \( c = 4 \) 2. **Find two numbers** that multiply to \( ac \) (where \( a \) is the coefficient of \( x^2 \) and \( c \) is the constant term): - \( ac = 3 \times 4 = 12 \) - Look for two numbers that multiply to 12 and add up to \( b \) (13). The numbers are \( 12 \) and \( 1 \). 3. **Rewrite the middle term** using the two numbers found: - \( 3x^2 + 12x + x + 4 \) 4. **Factor by grouping**: - Group the terms: \( (3x^2 + 12x) + (x + 4) \) - Factor out the common factor from each group: - \( 3x(x + 4) + 1(x + 4) \) 5. **Factor out the common binomial**: - \( (3x + 1)(x + 4) \) Thus, the factored form of \( 3x^2 + 13x + 4 \) is \( (3x + 1)(x + 4) \). Feel free to use this example as a guide for factoring similar quadratic expressions.
Transcribed Image Text:**Quadratic Factorization Exercise** --- **Problem Statement:** Factor the following quadratic expression: \( 3x^2 + 13x + 4 \) **Instructions:** To factor this quadratic expression, follow these steps: 1. **Identify the coefficients** in the quadratic equation of the form \( ax^2 + bx + c \): - \( a = 3 \) - \( b = 13 \) - \( c = 4 \) 2. **Find two numbers** that multiply to \( ac \) (where \( a \) is the coefficient of \( x^2 \) and \( c \) is the constant term): - \( ac = 3 \times 4 = 12 \) - Look for two numbers that multiply to 12 and add up to \( b \) (13). The numbers are \( 12 \) and \( 1 \). 3. **Rewrite the middle term** using the two numbers found: - \( 3x^2 + 12x + x + 4 \) 4. **Factor by grouping**: - Group the terms: \( (3x^2 + 12x) + (x + 4) \) - Factor out the common factor from each group: - \( 3x(x + 4) + 1(x + 4) \) 5. **Factor out the common binomial**: - \( (3x + 1)(x + 4) \) Thus, the factored form of \( 3x^2 + 13x + 4 \) is \( (3x + 1)(x + 4) \). Feel free to use this example as a guide for factoring similar quadratic expressions.
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