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Chapter 5: Problem 30

For which values of \(x\) is \(x^2-2 x+3\) greater than 3 ? A. \(1>x>3\) B. None C. \(-4x>2\)

Short Answer

Expert verified
B. None

Step by step solution

01

- Set Up Inequality

We need to determine for which values of \(x\) the expression \(x^2 - 2x + 3\) is greater than 3. Set up the inequality: \(x^2 - 2x + 3 > 3\).
02

- Simplify the Inequality

Subtract 3 from both sides to simplify the inequality: \(x^2 - 2x + 3 - 3 > 0\).
03

- Further Simplify

This simplifies to \(x^2 - 2x > 0\).
04

- Factor the Quadratic Expression

Factor the quadratic expression: \(x(x - 2) > 0\).
05

- Solve the Inequality

Solve the inequality: \(x(x - 2) > 0\). This inequality is positive when \(x < 0\) or \(x > 2\).
06

- Check the Options

Compare the solution \(x < 0\) or \(x > 2\) with the provided options: None of the given options exactly match the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Quadratic Inequalities
Quadratic inequalities involve expressions where the highest power of the variable is a square. For instance, the inequality in the exercise is defined by the quadratic expression \(x^2 - 2x + 3\). Solving quadratic inequalities often requires us to determine the range of values for the variable that satisfy the given condition, such as the expression being greater than or less than a certain value. It is crucial to set up the inequality accurately and transform it step by step.
Steps to Solving Quadratic Inequalities
To solve quadratic inequalities, follow a structured approach:

First, set the quadratic expression in the inequality form. For the given exercise, we start with \(x^2 - 2x + 3 > 3\).

Next, simplify the inequality by isolating terms on one side. Here, it becomes \(x^2 - 2x + 3 - 3 > 0\), simplifying to \(x^2 - 2x > 0\).

Remember: The goal is to find which values of \(x\) make the inequality true. This often involves factoring the quadratic expression, solving it, and analyzing the intervals where the product is positive or negative.
Factoring Quadratic Expressions
Factoring quadratic expressions is a key step in solving these inequalities. Once simplified, \(x^2 - 2x > 0\) is factored into \(x(x - 2) > 0\).

This step is vital because it transforms the quadratic expression into a product of simpler binomials. Factoring helps us see the critical points more clearly – the values of \(x\) where the expression equals zero. In this case, the critical points are \(x = 0\) and \(x = 2\).
It's essential to understand how to factor quadratics to break them down efficiently.
Solving and Interpreting Inequality Solutions
Once the quadratic expression is factored, solving the inequality involves determining the intervals where the product of factors is positive or negative. With \(x(x - 2) > 0\), the product is positive when both factors are either positive or negative.

We find that the expression is positive when:
  • \(x < 0\)
  • \(x > 2\)
The solution set becomes clear: \(x\) is either less than 0 or greater than 2. This final step includes comparing your solved intervals with given options and confirming the correct answer.

Note: Understanding how to test and interpret these intervals accurately is key to mastering quadratic inequalities.

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Most popular questions from this chapter

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