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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![**Example (d): Quadratic Inequality**
Consider the quadratic inequality:
\[x^2 - 4x + 4 \leq 0\]
To solve this inequality, we need to determine the values of \(x\) that satisfy the given condition. First, let's analyze the quadratic expression.
1. **Factorization:**
The expression \(x^2 - 4x + 4\) can be factored as:
\[(x - 2)^2\]
2. **Critical Points:**
The critical points occur where the expression equals zero:
\[(x - 2)^2 = 0\]
Solving this, we get:
\[x = 2\]
3. **Sign Analysis:**
The quadratic expression \((x - 2)^2\) represents a perfect square, which is always non-negative. Specifically, \((x - 2)^2 \ge 0\) for all real numbers \(x\).
4. **Inequality Analysis:**
Since the inequality is \(\leq 0\), we are interested in when the expression is less than or equal to zero. Analyze:
\[(x - 2)^2 \leq 0\]
Because \((x - 2)^2\) is always non-negative (i.e., it is zero or positive), the only time this inequality holds true is when the expression is exactly zero:
\[(x - 2)^2 = 0\]
As determined previously, this occurs at:
\[x = 2\]
Therefore, the solution to the quadratic inequality \(x^2 - 4x + 4 \leq 0\) is:
\[x = 2\]
This solution can be represented on a number line with a closed dot at \(x = 2\), indicating that \(2\) is included in the solution set. There are no intervals of \(x\) where the inequality holds, other than the single point \(x = 2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ae0660d-5301-4b30-8361-0f1bb7fa4959%2F22d97a91-b18d-408f-94fb-d80ae741b0eb%2Fdiiuw3_processed.png&w=3840&q=75)
Transcribed Image Text:**Example (d): Quadratic Inequality**
Consider the quadratic inequality:
\[x^2 - 4x + 4 \leq 0\]
To solve this inequality, we need to determine the values of \(x\) that satisfy the given condition. First, let's analyze the quadratic expression.
1. **Factorization:**
The expression \(x^2 - 4x + 4\) can be factored as:
\[(x - 2)^2\]
2. **Critical Points:**
The critical points occur where the expression equals zero:
\[(x - 2)^2 = 0\]
Solving this, we get:
\[x = 2\]
3. **Sign Analysis:**
The quadratic expression \((x - 2)^2\) represents a perfect square, which is always non-negative. Specifically, \((x - 2)^2 \ge 0\) for all real numbers \(x\).
4. **Inequality Analysis:**
Since the inequality is \(\leq 0\), we are interested in when the expression is less than or equal to zero. Analyze:
\[(x - 2)^2 \leq 0\]
Because \((x - 2)^2\) is always non-negative (i.e., it is zero or positive), the only time this inequality holds true is when the expression is exactly zero:
\[(x - 2)^2 = 0\]
As determined previously, this occurs at:
\[x = 2\]
Therefore, the solution to the quadratic inequality \(x^2 - 4x + 4 \leq 0\) is:
\[x = 2\]
This solution can be represented on a number line with a closed dot at \(x = 2\), indicating that \(2\) is included in the solution set. There are no intervals of \(x\) where the inequality holds, other than the single point \(x = 2\).
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