Problem 36 - Elementary and Intermediate Algebra

Question

Find all real numbers for which the rational expression is undefined. See Example 2. $$ \frac{x-20}{x^{2}+2 x-8} $$

Step-by-Step Solution

Answer

The expression is undefined for \(x = -4\) and \(x = 2\).

Step 1: Identify when the fraction is undefined

A rational expression becomes undefined when the denominator is equal to zero. Therefore, we must find values of \(x\) that make \(x^2 + 2x - 8 = 0\).

Step 2: Factor the quadratic denominator

We need to factor the quadratic expression \(x^2 + 2x - 8\). We look for two numbers that multiply to \(-8\) and add up to \(2\). The numbers \(4\) and \(-2\) fit this requirement, so the factorization is \((x + 4)(x - 2)\).

Step 3: Set each factor to zero

For the expression \((x + 4)(x - 2) = 0\), set each factor equal to zero: \(x + 4 = 0\) and \(x - 2 = 0\).

Step 4: Solve each equation

Solving the equation \(x + 4 = 0\) gives \(x = -4\). Solving the equation \(x - 2 = 0\) gives \(x = 2\). These are the values where the expression is undefined.

Key Concepts

Undefined Expressions

In mathematics, a rational expression is considered undefined when its denominator equals zero. This is because division by zero is not possible. For example, if you have a fraction \( \frac{a}{b} \), it becomes undefined at any point where \( b = 0 \). This is crucial for rational expressions as it helps determine where specific values of \( x \) cannot be used. To find such values, analyze the denominator of the given expression. Set the denominator equal to zero and solve for \( x \). Any solutions you find will indicate where the rational expression becomes undefined.

• Example: For the expression \( \frac{x-20}{x^2+2x-8} \), notice that it becomes undefined when \( x^2+2x-8 = 0 \).

Always ensure that you check the expression for undefined points in your calculations to avoid errors in solving and simplifying expressions.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, usually in the form \( ax^2 + bx + c = 0 \). Solving quadratic equations is a fundamental skill in algebra. Understanding the structure and solutions of these equations is essential when working with rational expressions, especially those that may be undefined.Quadratic equations can be solved using various methods:

• Factoring: Looks for two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add up to \( b \).
• Quadratic Formula: Given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the Square: A method where you manipulate the equation to form a perfect square trinomial.

For the given problem, we utilize factoring because it's often the simplest method for equations with small integer coefficients. This allows us to break down the equation \( x^2 + 2x - 8 = 0 \) into solvable linear factors.

Factoring Quadratics

Factoring is a method used to simplify quadratic expressions by breaking them down into their components. To factor a quadratic, you find two numbers that multiply to give you the product of \( a \times c \) in \( ax^2 + bx + c \) and add to give \( b \).In the exercise given, \( x^2 + 2x - 8 \) is a quadratic expression we need to factor. We look for two numbers that:

• Multiply to \(-8\), which is \( (1 \times -8) \).
• Add up to \( 2 \).

The numbers \( 4 \) and \(-2\) fit these criteria, as \( 4 \times -2 = -8 \) and \( 4 + (-2) = 2 \).Thus, the quadratic \( x^2 + 2x - 8 \) can be factored into \( (x + 4)(x - 2) \). Substituting this back into the rational expression, we can see that solving the equations \( x + 4 = 0 \) and \( x - 2 = 0 \) gives us the points where the expression becomes undefined, \( x = -4 \) and \( x = 2 \). This factorization process is not only important for finding when expressions are undefined but is also key in simplifying and solving quadratic and rational equations.