Jackie correctly solved this equation:
$\frac{4}{x-2}+\frac{1}{x^2-4}=\frac{1}{x+2}$
Which statement describes the solutions she found?
A. She found one valid solution and no extraneous solutions.
B. She found two valid solutions and no extraneous solutions.
C. She found one valid solution and one extraneous solution.
D. She found no valid solutions and two extraneous solutions.
Answer (1)
Factor the denominator: x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Multiply both sides by ( x − 2 ) ( x + 2 ) to get 4 ( x + 2 ) + 1 = x − 2 .
Simplify and solve for x : 4 x + 9 = x − 2 ⟹ 3 x = − 11 ⟹ x = − 3 11 .
Check for extraneous solutions: x = − 3 11 is not 2 or -2, so it is a valid solution. Thus, the answer is A .
Explanation
Analyzing the Problem We are given the equation x − 2 4 + x 2 − 4 1 = x + 2 1 and we need to determine the nature of the solutions Jackie found.
Factoring the Denominator First, we factor the denominator x 2 − 4 as ( x − 2 ) ( x + 2 ) . This allows us to rewrite the equation as x − 2 4 + ( x − 2 ) ( x + 2 ) 1 = x + 2 1 .
Eliminating Denominators To eliminate the denominators, we multiply both sides of the equation by ( x − 2 ) ( x + 2 ) . This gives us 4 ( x + 2 ) + 1 = ( x − 2 ) .
Solving for x Now, we simplify and solve for x :
4 x + 8 + 1 = x − 2 4 x + 9 = x − 2 3 x = − 11 x = − 3 11 .
Checking for Extraneous Solutions We need to check for extraneous solutions. Extraneous solutions occur when a solution makes any of the denominators in the original equation equal to zero. The denominators are x − 2 , x 2 − 4 , and x + 2 . These are zero when x = 2 or x = − 2 . Since our solution is x = − 3 11 , which is not equal to 2 or -2, it is not an extraneous solution.
Conclusion Since x = − 3 11 is the only solution and it is not extraneous, Jackie found one valid solution and no extraneous solutions.
Examples
When solving equations involving rational expressions, it's crucial to identify and exclude values that make the denominator zero. For instance, if you're designing a bridge and calculating stress distribution, a rational equation might arise. The solution must not lead to infinite stress (denominator equals zero), which would cause the bridge to collapse. Similarly, in electrical circuit analysis, you might encounter rational expressions for impedance. The solution must not result in zero impedance in certain parts of the circuit, as this could cause a short circuit or damage components. Therefore, understanding extraneous solutions is vital in engineering and physics to ensure the validity and safety of designs and calculations.
