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
Question
Solve the quadratic inequalities.
![The given inequality is:
\[ x^2 - 4x + 4 > 0 \]
This inequality represents a quadratic expression. To solve this inequality, we first need to analyze the quadratic equation \( x^2 - 4x + 4 = 0 \).
### Steps to solve the given quadratic inequality:
1. **Factor the Quadratic Expression:**
\[ x^2 - 4x + 4 = (x - 2)^2 \]
2. **Set the Quadratic Expression Equal to Zero:**
\[ (x - 2)^2 = 0 \]
Solve for \( x \):
\[ x - 2 = 0 \implies x = 2 \]
This indicates that the quadratic expression has a repeated root at \( x = 2 \).
3. **Examine the Sign of the Expression:**
Since \( (x - 2)^2 \) is a square term, it is always non-negative (\( \geq 0 \)). The equality holds at \( x = 2 \). For all other values of \( x \), \( (x - 2)^2 \) is positive.
4. **Solve the Inequality:**
For \( x^2 - 4x + 4 > 0 \), we need the expression to be positive:
\[ (x - 2)^2 > 0 \]
Since the square of any real number is non-negative and equal to zero only at \( x = 2 \), it is positive everywhere else. Therefore, the solution to the inequality is:
\[ x \neq 2 \]
### Final Answer:
\[ x \in (-\infty, 2) \cup (2, \infty) \]
### Explanation:
The quadratic inequality \( x^2 - 4x + 4 > 0 \) holds true for all real values of \( x \) except \( x = 2 \). The expression equals zero at \( x = 2 \) because it is a perfect square.
This information is essential for understanding quadratic inequalities and their solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ae0660d-5301-4b30-8361-0f1bb7fa4959%2Fa2e1a042-7001-4d0f-b1a6-e532899115af%2Faj8bbpv_processed.png&w=3840&q=75)
Transcribed Image Text:The given inequality is:
\[ x^2 - 4x + 4 > 0 \]
This inequality represents a quadratic expression. To solve this inequality, we first need to analyze the quadratic equation \( x^2 - 4x + 4 = 0 \).
### Steps to solve the given quadratic inequality:
1. **Factor the Quadratic Expression:**
\[ x^2 - 4x + 4 = (x - 2)^2 \]
2. **Set the Quadratic Expression Equal to Zero:**
\[ (x - 2)^2 = 0 \]
Solve for \( x \):
\[ x - 2 = 0 \implies x = 2 \]
This indicates that the quadratic expression has a repeated root at \( x = 2 \).
3. **Examine the Sign of the Expression:**
Since \( (x - 2)^2 \) is a square term, it is always non-negative (\( \geq 0 \)). The equality holds at \( x = 2 \). For all other values of \( x \), \( (x - 2)^2 \) is positive.
4. **Solve the Inequality:**
For \( x^2 - 4x + 4 > 0 \), we need the expression to be positive:
\[ (x - 2)^2 > 0 \]
Since the square of any real number is non-negative and equal to zero only at \( x = 2 \), it is positive everywhere else. Therefore, the solution to the inequality is:
\[ x \neq 2 \]
### Final Answer:
\[ x \in (-\infty, 2) \cup (2, \infty) \]
### Explanation:
The quadratic inequality \( x^2 - 4x + 4 > 0 \) holds true for all real values of \( x \) except \( x = 2 \). The expression equals zero at \( x = 2 \) because it is a perfect square.
This information is essential for understanding quadratic inequalities and their solutions.
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