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.

Answered: Image /qna-images/answer/a2e1a042-7001-4d0f-b1a6-e532899115af.jpg

Answered: ²-4x+4>0

²-4x+4>0

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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