Problem 28 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{x^{2}-4 x}{x^{2}+4} $$

Step-by-Step Solution

Answer

The expression is defined for all real numbers.

Step 1: Understand Expression Undefined

A rational expression is undefined where its denominator is equal to zero. The expression is given as \( \frac{x^{2}-4x}{x^{2}+4} \). To find where it is undefined, we need to focus on the denominator \( x^{2} + 4 \).

Step 2: Set Denominator to Zero

Set the denominator equal to zero to find the undefined points: \( x^{2} + 4 = 0 \).

Step 3: Solve the Equation

Solve the equation \( x^{2} + 4 = 0 \). This leads to \( x^{2} = -4 \). However, the square of any real number cannot be negative, so \( x^{2} = -4 \) has no real solutions.

Step 4: Conclusion on Real Numbers

Since there are no real numbers that satisfy the equation \( x^{2} = -4 \), the expression is never undefined for any real number.

Key Concepts

undefined expressions

In mathematics, a rational expression can sometimes become undefined. This occurs when the denominator of the expression is zero. A rational expression is essentially a fraction where the numerator and the denominator are both polynomials. If the denominator becomes zero, the whole expression is invalid or undefined. This is because division by zero is not possible in arithmetic. Always remember, any situation where the denominator is zero means the expression does not have a valid value.

denominator equals zero

You might wonder why a zero in the denominator causes so much trouble. In mathematics, division by zero does not produce a predictable outcome, so it’s considered undefined. To determine when a rational expression is undefined, we set its denominator equal to zero and solve for the variable. For instance, in the expression \( \frac{x^2 - 4x}{x^2 + 4} \), you would set \( x^2 + 4 = 0 \) to find any points where the denominator equals zero. This helps in determining potential undefined points.

real numbers

Real numbers are the numbers that can be found on the number line. This includes both rational numbers (like integers and fractions) and irrational numbers. They do not include imaginary or complex numbers. In context of rational expressions, we are often concerned with finding real number solutions. However, some equations have solutions that are not real numbers. For example, the equation \( x^2 = -4 \) does not have real number solutions because no square of a real number is negative. Thus, sometimes the expression remains defined for all real numbers, as seen in our previous example.

solving equations

Solving equations often involves finding the value of the variable that makes an equation true. When dealing with rational expressions, part of solving involves ensuring the denominator does not equal zero. We begin by setting the denominator equal to zero. If the resulting equation has real solutions, those are the points where the original expression is undefined. In our given expression \( \frac{x^2 - 4x}{x^2 + 4} \), we solve \( x^2 + 4 = 0 \) and find no real number solutions. Thus, in this case, solving the equation confirms no real numbers make the denominator zero, keeping the expression defined across all real numbers.