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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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
![### Solving Quadratic Equations
#### Problem Statement:
Find the solution(s) to the following quadratic equation:
\[
(x - 4)(2x - 3) = 0
\]
Solutions:
\[
x = 4, \quad \frac{3}{2}
\]
If you have any questions, feel free to click on "Edit my response" for further clarifications or to try solving similar problems.
#### Step-by-Step Explanation:
To solve the quadratic equation \((x - 4)(2x - 3) = 0\), we use the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for \(x\).
1. **First Factor:**
\[
x - 4 = 0
\]
Adding 4 to both sides:
\[
x = 4
\]
2. **Second Factor:**
\[
2x - 3 = 0
\]
Adding 3 to both sides:
\[
2x = 3
\]
Then, dividing both sides by 2:
\[
x = \frac{3}{2}
\]
Thus, the solutions are \(x = 4\) and \(x = \frac{3}{2}\).
#### Review:
To ensure the solutions are correct, we substitute them back into the original equation:
- For \(x = 4\):
\[
(4 - 4)(2(4) - 3) = 0 \cdot 5 = 0
\]
- For \(x = \frac{3}{2}\):
\[
\left( \frac{3}{2} - 4 \right)\left(2 \cdot \frac{3}{2} - 3\right) = \left( \frac{-5}{2} \right) \left(3 - 3\right) = \left( \frac{-5}{2} \right) \cdot 0 = 0
\]
Both solutions satisfy the original equation.
Feel free to experiment with different forms of quadratic equations and test your understanding with more examples!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb6a3e0a-0e9e-4eb9-b3f8-cac67ff0b1b6%2F7122419b-19e4-4369-b13a-a955887538fd%2F1netug.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Quadratic Equations
#### Problem Statement:
Find the solution(s) to the following quadratic equation:
\[
(x - 4)(2x - 3) = 0
\]
Solutions:
\[
x = 4, \quad \frac{3}{2}
\]
If you have any questions, feel free to click on "Edit my response" for further clarifications or to try solving similar problems.
#### Step-by-Step Explanation:
To solve the quadratic equation \((x - 4)(2x - 3) = 0\), we use the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor to zero and solve for \(x\).
1. **First Factor:**
\[
x - 4 = 0
\]
Adding 4 to both sides:
\[
x = 4
\]
2. **Second Factor:**
\[
2x - 3 = 0
\]
Adding 3 to both sides:
\[
2x = 3
\]
Then, dividing both sides by 2:
\[
x = \frac{3}{2}
\]
Thus, the solutions are \(x = 4\) and \(x = \frac{3}{2}\).
#### Review:
To ensure the solutions are correct, we substitute them back into the original equation:
- For \(x = 4\):
\[
(4 - 4)(2(4) - 3) = 0 \cdot 5 = 0
\]
- For \(x = \frac{3}{2}\):
\[
\left( \frac{3}{2} - 4 \right)\left(2 \cdot \frac{3}{2} - 3\right) = \left( \frac{-5}{2} \right) \left(3 - 3\right) = \left( \frac{-5}{2} \right) \cdot 0 = 0
\]
Both solutions satisfy the original equation.
Feel free to experiment with different forms of quadratic equations and test your understanding with more examples!
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