Problem 35 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{15}{x^{2}+x-2} $$

Step-by-Step Solution

Answer

The rational expression is undefined for \(x = -2\) and \(x = 1\).

Step 1: Identify the Denominator

To find when the rational expression is undefined, we need to focus on the denominator. The expression is undefined when the denominator is equal to zero. In this case, the denominator is \(x^2 + x - 2\).

Step 2: Set the Denominator to Zero

Set the denominator equal to zero to find the values of \(x\) that make the expression undefined: \(x^2 + x - 2 = 0\).

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \(x^2 + x - 2 = 0\). We can factor this equation to find the values of \(x\).

Step 4: Factor the Quadratic

Look for factors of \(-2\) that add to \(1\). These numbers are \(2\) and \(-1\). So, we can factor the quadratic as \((x + 2)(x - 1) = 0\).

Step 5: Find the Roots

Set each factor equal to zero and solve for \(x\): 1. \(x + 2 = 0\) gives \(x = -2\).2. \(x - 1 = 0\) gives \(x = 1\).

Step 6: Identify the Undefined Points

The expression is undefined for the values of \(x\) that make the denominator zero. Hence, the expression is undefined at \(x = -2\) and \(x = 1\).

Key Concepts

Undefined Rational Expressions

Rational expressions become undefined when the denominator equals zero. Imagine trying to divide something by nothing, which is essentially what is happening when the denominator of a rational expression is zero. This makes the expression undefined at those particular points.

To find when a rational expression, like \( \frac{15}{x^{2}+x-2} \), is undefined, we need to focus on the denominator. We set it equal to zero: \( x^2 + x - 2 = 0 \). This quadratic equation will show us the values of \( x \) that cause the denominator to become zero. Solving this quadratic is crucial to determine these undefined points. As a result, our main task is to solve the equation, as these solutions will represent where the expression fails to exist.

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our example, the equation \( x^2 + x - 2 = 0 \) is a straightforward quadratic. These equations often appear in algebra, and solving them is a key skill.

Quadratics can be solved using:

• Factoring (useful for simple factorable equations)
• Quadratic formula (a must-know when other methods fail)
• Completion of the square (for those comfortable with transforming expressions)

For this particular equation, factoring is the most straightforward approach due to the simple nature of \( x^2 + x - 2 \). Successfully solving these equations helps us find where a rational expression like ours becomes undefined.

Factoring Quadratics

Factoring is a method of breaking down a quadratic equation into simpler expressions (factors) that can be easily solved. In the equation \( x^2 + x - 2 = 0 \), factoring is particularly straightforward. We seek two numbers that multiply to \( -2 \) (the constant term) and add up to \( 1 \) (the coefficient of \( x \)). These numbers are \( 2 \) and \( -1 \).

Thus, we can rewrite the quadratic as \( (x + 2)(x - 1) = 0 \). By setting each factor equal to zero—\( x + 2 = 0 \) and \( x - 1 = 0 \)—we find the roots: \( x = -2 \) and \( x = 1 \).

These roots are incredibly important because they give us the exact points where the original expression is undefined. Understanding factoring is a valuable skill as it can simplify seemingly complex equations and help us solve them efficiently.