Problem 4 - Elementary Algebra

Question

Find the domain of each of the following rational expressions. $$ \frac{5 a+2}{a^{2}+6 a+8} $$

Step-by-Step Solution

Answer

Answer: The domain of the rational expression is all real numbers except for \(a=-2\) and \(a=-4\), represented as \(\{a \in \mathbb{R} \ | \ a \neq -2 \text{ and } a \neq -4 \}\).

Step 1: Set the Denominator Equal to Zero

The first step in finding the domain is to set the denominator equal to zero and solve for the variable. In this case, the denominator is \(a^{2}+6a+8\). $$ a^{2}+6a+8 = 0 $$

Step 2: Solve the Quadratic Equation

Next, we need to solve the resulting quadratic equation. We can factor this equation to find the roots. $$ (a+2)(a+4) = 0 $$ Now, we can set each factor equal to zero and solve for \(a\). $$ a+2=0 \Rightarrow a=-2 $$ $$ a+4=0 \Rightarrow a=-4 $$

Step 3: Determine the Domain

We found that \(a=-2\) and \(a=-4\) make the denominator equal to zero. To find the domain, we exclude these values from the set of all real numbers. The domain of the rational expression is: $$ Domain = \{a \in \mathbb{R} \ | \ a \neq -2 \text{ and } a \neq -4 \} $$

Key Concepts

Quadratic Equations

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \), where \( x \) represents an unknown, and \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Quadratic equations are crucial because they illustrate the concept of expressions that curve rather than form a straight line. A quadratic equation can have zero, one, or two solutions. These solutions are found at the points where the quadratic expression equals zero, often referred to as the roots or zeros of the equation.
To find these solutions, we often use factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Each method is useful in different situations, but a clear understanding of the properties of quadratic expressions can guide you in deciding which method to use effectively.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions. It's like reverse multiplication. In the context of quadratic equations like \( a^2 + 6a + 8 \), factoring helps us to find the roots or solutions when the quadratic equals zero.
Let's illustrate factoring with \( a^2 + 6a + 8 \). You seek two numbers that multiply to 8 (the constant term) and add to 6 (the coefficient of \( a \)). These numbers are 2 and 4. Thus, the expression \( a^2 + 6a + 8 \) factors into \((a + 2)(a + 4)\).
Factoring provides a straightforward way to find solutions: set each factor to zero, yielding \( a + 2 = 0 \) and \( a + 4 = 0 \). Hence, the solutions \( a = -2 \) and \( a = -4 \) emerge. Understanding factoring is vital not only for solving quadratic equations but also simplifies expressions and supports evaluating rational expressions.

Rational Expressions

Rational expressions are fractions that include polynomials both in the numerator and the denominator, such as \( \frac{5a + 2}{a^2 + 6a + 8} \). In mathematics, understanding the domain of rational expressions is key. The domain encompasses all the possible values of the variable for which the expression is defined. Since division by zero is undefined, one crucial task is to identify and exclude any values that make the denominator zero.
For example, to determine the domain of \( \frac{5a + 2}{a^2 + 6a + 8} \), we first find when the denominator equals zero by solving the quadratic \( a^2 + 6a + 8 = 0 \). Upon factoring, as shown earlier, this yields \( a = -2 \) and \( a = -4 \), meaning these values make the expression undefined. Therefore, all real numbers except \( a = -2 \) and \( a = -4 \) are included in the domain. So, we express the domain as \( \{ a \in \mathbb{R} \ | \ a eq -2 \text{ and } a eq -4 \} \). Mastering the examination of these expressions ensures that you retrieve all permissible values while avoiding mathematical errors associated with division by zero.