Problem 86 - Elementary Algebra

Question

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{5}{x^{2}+6} $$

Step-by-Step Solution

Answer

The expression is always defined for all real values of \(x\).

Step 1: Understand the Problem

To determine when the given expression is undefined, identify where the denominator equals zero. The expression is undefined whenever the denominator is zero.

Step 2: Set Denominator to Zero

Set the denominator equal to zero and solve for the variable: \( x^{2} + 6 = 0 \)

Step 3: Solve the Equation

Isolate the variable term: \( x^{2} = -6 \). Now, take the square root on both sides to solve for \(x\):\( x = \pm \sqrt{-6} \)

Step 4: Interpret the Result

Note that the square root of a negative number is not a real number. Therefore, there are no real values of \(x\) that make the denominator zero.

Key Concepts

Denominator Equals Zero

To understand when a mathematical expression with a denominator is undefined, we need to focus on the denominator itself. Specifically, the expression becomes undefined when the denominator equals zero. This is because division by zero is not possible in mathematics.

In the exercise given, the expression is \( \frac{5}{x^2 + 6} \). To determine when this expression is undefined, we need to find the values of \( x \) that make the denominator zero:

• Set the denominator equal to zero: \( x^2 + 6 = 0 \).
• Solve for \( x \).

Once we find such values, the original expression is undefined at those points.

Solving Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants. To solve these equations, multiple methods can be used, such as factoring, completing the square, or using the quadratic formula.

For our exercise, after setting the denominator to zero, we get the quadratic equation \( x^2 + 6 = 0 \). Let's solve it:

• Isolate the quadratic term: \( x^2 = -6 \).
• Take the square root of both sides: \( x = \pm \sqrt{-6} \).

Now we face the issue of taking the square root of a negative number, which leads us to the concept of imaginary numbers.

Imaginary Numbers

When solving the quadratic equation \( x^2 = -6 \), we need to understand imaginary numbers. In mathematics, the imaginary unit \( i \) is defined as \( \sqrt{-1} \). Using this definition, we can express the square root of any negative number.

For our equation:

• We find \( \sqrt{-6} = \sqrt{6}i \).
• Thus, \( x = \pm i\sqrt{6} \).

This means there are no real values of \( x \) that make the denominator zero, but rather complex numbers involving the imaginary unit \( i \).

In conclusion, for the given expression \( \frac{5}{x^2 + 6} \), there are no real values of \( x \) that make the expression undefined. However, it is important to understand the role of imaginary numbers in solving equations with negative square roots.