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,...
Related questions
Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![# Factor and Zero Product Property 8-112 b
## Show AND explain your work using details specific to this problem.
### Solve for x using the Zero Product Property.
Given equation:
\[ 0 = 6x^2 - 23x + 20 \]
### Steps to solve:
1. **Factor the quadratic equation:**
To factor the quadratic equation \(6x^2 - 23x + 20\), look for two binomials such that their product gives back the quadratic equation.
2. **Find the factors:**
The equation can be factored into two binomials, which involves finding two numbers whose product is the constant term (20*6 = 120) and whose sum is the coefficient of the middle term (-23).
3. **Rewrite the middle term using the found factors:**
Rewrite -23x using the found factors (e.g., -15x and -8x):
\[
6x^2 - 15x - 8x + 20
\]
4. **Factor by grouping:**
Group the terms:
\[
(6x^2 - 15x) - (8x - 20)
\]
Factor out the common terms from each group:
\[
3x(2x - 5) - 4(2x - 5)
\]
Factor out the common binomial:
\[
(3x - 4)(2x - 5)
\]
5. **Apply the Zero Product Property:**
Set each factor equal to zero:
\[
3x - 4 = 0 \quad \text{or} \quad 2x - 5 = 0
\]
6. **Solve for x:**
Solve each equation separately:
\[
3x - 4 = 0 \implies 3x = 4 \implies x = \frac{4}{3}
\]
\[
2x - 5 = 0 \implies 2x = 5 \implies x = \frac{5}{2}
\]
### Solution:
The roots of the equation are:
\[
x = \frac{4}{3} \quad](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90d1ba02-23bf-4f72-abf7-6dbb2536cbf6%2F339251e1-d03e-4d3d-9528-5e4f384a1253%2F1l5hs19_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Factor and Zero Product Property 8-112 b
## Show AND explain your work using details specific to this problem.
### Solve for x using the Zero Product Property.
Given equation:
\[ 0 = 6x^2 - 23x + 20 \]
### Steps to solve:
1. **Factor the quadratic equation:**
To factor the quadratic equation \(6x^2 - 23x + 20\), look for two binomials such that their product gives back the quadratic equation.
2. **Find the factors:**
The equation can be factored into two binomials, which involves finding two numbers whose product is the constant term (20*6 = 120) and whose sum is the coefficient of the middle term (-23).
3. **Rewrite the middle term using the found factors:**
Rewrite -23x using the found factors (e.g., -15x and -8x):
\[
6x^2 - 15x - 8x + 20
\]
4. **Factor by grouping:**
Group the terms:
\[
(6x^2 - 15x) - (8x - 20)
\]
Factor out the common terms from each group:
\[
3x(2x - 5) - 4(2x - 5)
\]
Factor out the common binomial:
\[
(3x - 4)(2x - 5)
\]
5. **Apply the Zero Product Property:**
Set each factor equal to zero:
\[
3x - 4 = 0 \quad \text{or} \quad 2x - 5 = 0
\]
6. **Solve for x:**
Solve each equation separately:
\[
3x - 4 = 0 \implies 3x = 4 \implies x = \frac{4}{3}
\]
\[
2x - 5 = 0 \implies 2x = 5 \implies x = \frac{5}{2}
\]
### Solution:
The roots of the equation are:
\[
x = \frac{4}{3} \quad
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education