Chapter 11.5, Problem 5P

To determine

To solve the equation $\frac{x-4}{x^2-4}=\frac{-2}{x-2}$ and check the solution.

Answer to Problem 5P

  $x=0$

Explanation of Solution

Given:

Given equation

  $\frac{x-4}{x^2-4}=\frac{-2}{x-2}$

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
1. Multiply both sides of equations by LCD(least common denominator) of all denominators.
2. Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

In order to solve the equation $\frac{x-4}{x^2-4}=\frac{-2}{x-2}$ , cross multiply the expressions and simplify further as shown below

  $\begin{array}{c}\frac{x-4}{x^2-4}=\frac{-2}{x-2}\\ \left(x-4\right)\cdot \left(x-2\right)=-2\cdot \left(x^2-4\right)\\ x^2-2x-4x+8=-2x^2+8\\ 3x^2-6x=0\\ 3x\left(x-2\right)=0\\ 3x=0\text{or}x-2=0\\ x=0\text{or}x=2\end{array}$

Thus, the solutions of the given equation are $x=0,2$ .

Now to check the solution, substitute $x=0$ in the given equation and check whether left hand side value is equal to the right hand side value or no. So, it gives

  $\begin{array}{c}\frac{0-4}{{\left(0\right)}^2-4}=\frac{-2}{0-2}\\ \frac{-4}{-4}=\frac{-2}{-2}\\ 1=1\text{True Statement}\end{array}$

Again to check the other solution, substitute $x=2$ in the given equation and check whether left hand side value is equal to the right hand side value or no. So, it gives

  $\begin{array}{c}\frac{2-4}{{\left(2\right)}^2-4}=\frac{-2}{2-2}\\ \frac{-2}{4-4}=\frac{-2}{0}\\ \text{Undefined}\end{array}$

So, $x=2$ is an extraneous solution.

Thus, the only solution of the given equation is $x=0$ .