Problem 13 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 14 , find all the values of \(x\) for which the number is real. $$ \sqrt{x^{2}-5 x+4} $$

Step-by-Step Solution

Answer

The values of \( x \) for which the number is real are \( x \leq 1 \) or \( x \geq 4 \).

Step 1 - Identify the expression under the square root

The given expression is \ \( \sqrt{x^{2} - 5x + 4} \).

Step 2 - Set the expression under the square root to be greater than or equal to zero

For the square root to be real, the expression under the square root must be non-negative. Therefore, we need to solve the inequality: \ \( x^{2} - 5x + 4 \geq 0 \).

Step 3 - Factor the quadratic expression

We factor the quadratic expression \( x^{2} - 5x + 4 \) to find its roots. \ \( x^{2} - 5x + 4 = (x - 1)(x - 4) \).

Step 4 - Find the critical points

Set each factor to zero to find the critical points. \ \( x - 1 = 0 \rightarrow x = 1 \) and \ \( x - 4 = 0 \rightarrow x = 4 \).

Step 5 - Determine the intervals to test

The critical points divide the number line into three intervals: \ 1. \( x < 1 \) \ 2. \( 1 \leq x \leq 4 \) \ 3. \( x > 4 \). Test each interval to see where the expression \( (x - 1)(x - 4) \geq 0 \) holds true.

Step 6 - Test the intervals

1. For \( x < 1 \): both factors \( x - 1 \) and \( x - 4 \) are negative, making their product positive. \ 2. For \( 1 \leq x \leq 4 \): one factor is negative and one is positive, making their product negative. \ 3. For \( x > 4 \): both factors are positive, making their product positive.

Step 7 - Summarize the solution

The intervals where the quadratic expression is non-negative are: \ \( x \leq 1 \) or \( x \geq 4 \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically written as \(ax^2 + bx + c = 0\). They are significant because they frequently appear in various areas of mathematics, physics, and engineering.

To solve a quadratic equation, you can use several methods such as:

• Factoring: This involves rewriting the original equation as a product of two linear factors. For example, in our exercise, \(x^2 - 5x + 4\) was factored into \((x - 1)(x - 4)\).
• Quadratic Formula: This is a universal method given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This method works for any quadratic equation.
• Completing the Square: This method involves transforming the equation into a perfect square trinomial.

The goal of solving the quadratic equation in this exercise is to find the roots or critical points (1 and 4). These points help us determine intervals for further testing in the inequality phase.

Inequalities

In mathematics, inequalities are used to compare two values or expressions. An inequality may be strict (e.g., \(a < b\)) or non-strict (e.g., \(a \leq b\)).

• Quadratic Inequalities: These involve quadratic expressions and are solved by finding the intervals where the expression satisfies the inequality. For example, in our exercise, we need \(x^2 - 5x + 4 \geq 0\).
• Interpreting Inequalities: When solving inequalities, always pay attention to the direction of the inequality sign (greater than or equal to). It determines where to shade on the number line.

Understanding how to handle these differences is crucial. Here, the goal was to find the values of \(x\) where the quadratic expression was non-negative. Hence, we end up with the intervals where \(x \leq 1\) or \(x \geq 4\).

Interval Testing

Interval testing is a method used to determine where an inequality holds true by breaking the number line into intervals bounded by the critical points.

Let's break down the steps involved in interval testing:

• Identify Critical Points: First, find where the expression changes sign by solving the related equation. In our example, the critical points were found at \(x = 1\) and \(x = 4\).
• Divide the Number Line: Use the critical points to divide the number line into intervals. Here, the intervals are: \(x < 1\), \(1 \leq x \leq 4\), and \(x > 4\).
• Test Each Interval: Select a test point from each interval and substitute it back into the inequality to check if it holds true. For instance, for \(x < 1\), any value less than 1 can be checked to see that both factors are negative, resulting in a positive product.

The result of interval testing helps identify the intervals in which the original inequality holds. In this exercise, the intervals where the expression is non-negative are \(x \leq 1\) and \(x \geq 4\). This systematic approach ensures that students can accurately understand and solve inequalities involving quadratics.