Problem 123 - College Algebra Essentials

Question

In Exercises \(123-124,\) list all numbers that must be excluded from the domain of each rational expression. $$ \frac{3}{2 x^{2}+4 x-9} $$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain are \(x_1=\frac{-4+\sqrt{88}}{4}\) and \(x_2=\frac{-4-\sqrt{88}}{4}\).

Step 1: Find the denominator

The denominator of the given rational expression is \(2x^{2}+4x-9\). We need to find the values of \(x\) that will make this denominator zero.

Step 2: Set the denominator equal to zero

To find those values, we solve the equation \(2x^{2}+4x-9 = 0\).

Step 3: Solve the equation

Applying the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a=2\), \(b=4\), and \(c=-9\), we find that \(x=\frac{-4\pm\sqrt{4^{2}-4*2*-9}}{2*2}= \frac{-4\pm\sqrt{16+72}}{4}=\frac{-4\pm\sqrt{88}}{4}\). Therefore, \(x_1=\frac{-4+\sqrt{88}}{4}\) and \(x_2=\frac{-4-\sqrt{88}}{4}\) are the roots of the equation.

Step 4: Identify the excluded numbers

The excluded numbers from the domain of \(\frac{3}{2x^{2}+4x-9}\) are \(x_1=\frac{-4+\sqrt{88}}{4}\) and \(x_2=\frac{-4-\sqrt{88}}{4}\). These are the values of \(x\) that will make the denominator of the rational expression zero.

Key Concepts

Domain Exclusion

When dealing with rational expressions, certain values can make the expression undefined. These occur when the denominator is equal to zero. This leads to what is called **domain exclusion**. Unlike polynomial expressions, rational expressions have restrictions on their domain. To find the excluded values

• Set the denominator equal to zero.
• Solve the resulting equation for the variable.

These solutions are the values that must be excluded from the domain because they make the expression undefined. It is crucial to determine these values to correctly define where the function is valid.
In our example, for the expression \(\frac{3}{2x^{2}+4x-9}\), we must exclude the roots of the equation \(2x^{2}+4x-9=0\), as these values make the denominator zero, rendering the expression undefined.

Quadratic Formula

The quadratic formula is a vital tool in algebra for solving equations of the form \(ax^2 + bx + c = 0\). It provides the solutions or roots of the quadratic equation. The formula is expressed as\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how you can use it:

• Identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Substitute these values into the formula.
• Calculate the value under the square root (known as the discriminant).
• Determine the possible solutions for \(x\) by simplifying the expression.

For the problem given, using \(a=2\), \(b=4\), and \(c=-9\), you apply the quadratic formula to find the domain exclusions. This process will give you two potential solutions, which determine the values to exclude from the domain.

Roots

Roots are solutions to equations, specifically where the expression equals zero. In the context of a quadratic equation, these roots are the values of \(x\) that satisfy the equation
For an expression such as \(2x^2 + 4x - 9 = 0\), solving for the roots involves a few steps:

• Use the quadratic formula to solve for \(x\).
• Calculate each part of the formula carefully to ensure accuracy.
• The solutions for \(x\) are your roots, which may be real, repeated, or complex depending on the discriminant \((b^2 - 4ac)\).

In this example, the discriminant is positive, indicating two distinct real roots.
These roots represent the values of \(x\) where the denominator equals zero, making them the values that you exclude from the rational expression's domain.

Denominator

The denominator is a crucial component of rational expressions. It appears beneath the fraction bar and dictates whether the expression is valid or not. A rational expression becomes undefined when the denominator is zero, which is why finding excluded values is essential.
For our original expression \(\frac{3}{2x^{2}+4x-9}\), the denominator is \(2x^{2}+4x-9\). This quadratic expression must not be zero, so we determine the roots of the equation by setting up \(2x^{2}+4x-9=0\).

• Factor the quadratic, if possible, or use the quadratic formula to identify the roots.
• These roots are the critical values that, when substituted into the denominator, would result in zero.

Understanding the denominator's composition helps in identifying domain exclusions and ensuring the expression is effectively solved.