Problem 71 - Algebra for College Students

Question

Why is the rational expression \(\frac{x+3}{x^{2}-4}\) undefined for \(x=2\) and \(x=-2\) but defined for \(x=-3 ?\)

Step-by-Step Solution

Answer

The expression is undefined when the denominator is zero (at \(x=2\) and \(x=-2\)), defined when it is non-zero (at \(x=-3\)).

Step 1: Identify Conditions for Undefined Expressions

A rational expression is undefined when its denominator equals zero. Therefore, identify the values of \(x\) that make the denominator zero.

Step 2: Set Denominator to Zero and Solve

Given the expression \(\frac{x+3}{x^{2}-4}\), the denominator is \(x^{2}-4\). Set this equal to zero: \(x^{2}-4=0\).

Step 3: Solve the Quadratic Equation

Solve the equation \(x^{2}-4=0\) for \(x\): Factor it as \((x-2)(x+2)=0\). Thus, \(x=2\) and \(x=-2\) are the solutions.

Step 4: State the Undefined Conditions

The expression is undefined for \(x=2\) and \(x=-2\) because those values make the denominator zero.

Step 5: Evaluate at \(x=-3\)

Evaluate the expression when \(x=-3\): The denominator becomes \((-3)^{2}-4=9-4=5\), which is non-zero.

Step 6: Conclusion on Defined or Undefined Status

Since the denominator is non-zero when \(x=-3\), the expression is defined at this point.

Key Concepts

Undefined Expressions

Rational expressions can sometimes be tricky, especially when they become undefined. An undefined expression occurs when the denominator of a rational expression equals zero. Why is this important? Because any number divided by zero is undefined in mathematics! In our example, we are given the rational expression \(\frac{x+3}{x^2-4}\). To find when it's undefined, we set the denominator equal to zero: \(x^2-4=0\). When we solve this, we find that the expression is undefined for \(x=2\) and \(x=-2\) because these values make the denominator zero and, therefore, the entire expression undefined. It is key to remember:

• Undefined expressions occur when the denominator is zero.
• Setting the denominator to zero helps identify problematic values of \(x\).
• Rational expressions are only defined when the denominator is non-zero.

Quadratic Equations

Quadratic equations are invaluable tools in algebra. They usually appear in the form \(ax^2 + bx + c = 0\). Solving these equations often helps determine when an expression, like our rational one, is undefined. In the problem, you encounter the equation \(x^2-4=0\). This is a simplified quadratic equation with no \(b\) term. Here's how you solve it:

• Identify the equation to solve: \(x^2-4=0\).
• Rearrange if needed, but here it is ready to go.
• Solve by factoring, completing the square, or using the quadratic formula.

For \(x^2-4=0\), factoring is straightforward: notice it becomes \((x-2)(x+2)=0\). From this factored form, solving for \(x\) gives \(x=2\) and \(x=-2\). Both values cause our denominator in the original expression to zero out, making them the key to identifying undefined points. Remember:

• Simplifying quadratic equations can reveal when expressions are undefined.
• Factoring is a common method to solve simple quadratics.

Factoring Polynomials

Factoring polynomials is a crucial skill in tackling algebraic problems, including identifying when expressions are undefined. It's the process of breaking down a polynomial into simpler factors that, when multiplied together, give you the original polynomial. In the equation \(x^2-4\), we factor it as \((x-2)(x+2)\), which effortlessly shows us the values of \(x\) that result in the polynomial equating to zero.

Key points to master factoring include:

• Recognize patterns – some quadratics can fall into known identities like difference of squares.
• Break down the polynomial using these patterns, if possible.
• Test your factors by multiplying them back to see if they reconstruct the original polynomial.

For \(x^2 - 4\), recognize it as a difference of squares: \(x^2\) is a square, and so is 4 (\(2^2\)). Thus, it factors to \((x-2)(x+2)\). Factorization reveals \(x=2\) and \(x=-2\) as values that make the expression undefined when these are the denominator. These strategies not only help with undefined expressions but also enhance your overall algebraic dexterity.