Q. 7.19 - Intermediate Algebra

Question

Find the domain of $R\left(x\right)=\frac{2x^2-10x}{4x^2-16x-20}$.

Step-by-Step Solution

Answer

The domain of $R\left(x\right)$ is all real numbers, where $x\ne 5\text{ and }x\ne -1.$

Step 1 . Given information

The given expression is$R\left(x\right)=\frac{2x^2-10x}{4x^2-16x-20}$

Step 2 . Concept used

The domain of the rational function is all real numbers except the values that make the denominator equal to zero.

Procedure to find the domain:

• Set the denominator equal to zero.
• Solve the equation.
• The domain is all real numbers except the values found in the previous step.

Step 3 . Find the domain

$R\left(x\right)=\frac{2x^2-10x}{4x^2-16x-20}$

Set the denominator to zero and solve

$$\begin{gathered} 4x^2-16x-20=0 \\ 4x^2-20x+4x-20=0 \text{ Expand the middle term} \\ 4x(x-5)+4(x-5)=0 \text{ write out common elements} \\ (x-5)(4x+4)=0 \text{ write out common term} \\ x-5=0 \text{or }4x+4=0 \text{ Solve for }x \\ x=5 \text{ or } x=-1 \end{gathered}$$

Hence, the domain is all real numbers except at $x=5 , -1.$