Problem 26 - Algebra 1

Question

For what values of the variable is the rational expression undefined? $$\frac{4}{x^{2}-1}$$

Step-by-Step Solution

Answer

The values of x for which the rational expression \( \frac{4}{x^{2}-1} \) is undefined are \( x = 1 \) and \( x = -1 \).

Step 1: Identify the Denominator

The denominator of the given rational expression: \( \frac{4}{x^{2}-1} \) is \( x^{2}-1 \). This expression will be undefined when its denominator is zero.

Step 2: Set the Denominator Equal to Zero

To find the roots of the denominator, set \( x^{2}-1 = 0 \). This will give the values of \( x \) for which the rational expression is undefined.

Step 3: Solve the Equation

Solving the equation \( x^{2}-1 = 0 \), we get \( x^{2} = 1 \). Thus, \( x = \sqrt{1} \) & \( x = -\sqrt{1} \) Hence, \( x = 1 \) & \( x = -1 \).

Key Concepts

Undefined Values

The concept of undefined values is crucial in understanding rational expressions. A rational expression is undefined when the denominator equals zero because division by zero is not possible in mathematics. This results in a value that cannot be determined or interpreted.
For instance, consider the expression \(\frac{4}{x^2-1}\). To find out when this expression is undefined, we look for values that make the denominator \(x^2-1\) equal to zero.
Here's a simple method to determine undefined values:

• Identify the denominator of the rational expression.
• Set the denominator equal to zero.
• Solve the resulting equation to find the values that cause the expression to be undefined.

This allows you to safely determine when the expression cannot be evaluated.

Denominator

Understanding the role of the denominator in rational expressions is fundamental. The denominator is the part of a fraction situated below the division line, and it indicates the divisor in the expression.
In the rational expression \(\frac{4}{x^2-1}\), the denominator is \(x^2-1\). Mathematical operations require that the denominator never equals zero since it represents the divisor. When it becomes zero, the expression loses its meaning.
To manage this, follow these steps:

• Identify the denominator in the rational expression.
• Determine when the denominator equals zero, as this indicates undefined status.
• Investigate and solve the equation that results from setting the denominator to zero.

Comprehending the denominator's role is essential in avoiding undefined results in calculations.

Quadratic Equations

Quadratic equations frequently appear in algebra, including solving for when certain expressions are undefined. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\).
In the context of rational expressions like \(\frac{4}{x^2-1}\), solving the equation \(x^2-1 = 0\) plays a critical role. This is a simple quadratic equation, as it only involves \(x^2\) and a constant.
Here's how to handle solving these:

• Isolate the quadratic term by setting the equation to \(x^2 = 1\).
• Solve for \(x\) by taking the square root of both sides, giving \(x = \sqrt{1}\) and \(x = -\sqrt{1}\).
• Conclude with \(x = 1\) and \(x = -1\), which make the denominator zero, rendering the expression undefined at these points.

Understanding quadratic equations allows you to solve for undefined values in rational expressions effectively.