Complete the work shown to find a possible solution of the equation.
$\begin{array}{l}
(x-5)^{\frac{1}{2}}+5=2 \\
(x-5)^{\frac{1}{2}}=-3 \\
{\left[(x-5)^{\frac{1}{2}}\right]^2=(-3)^2}
\end{array}$
A possible solution of the equation is $\square$
Answer (2)
Simplify the equation [( x − 5 ) 2 1 ] 2 = ( − 3 ) 2 to get x − 5 = 9 .
Isolate x by adding 5 to both sides: x = 9 + 5 = 14 .
Check if x = 14 is a valid solution in the original equation. It is not, but the question asks for a possible solution based on the work shown.
A possible solution of the equation is 14 .
Explanation
Analyzing the Problem We are given the equation ( x − 5 ) 2 1 + 5 = 2 and the steps taken to solve it:
( x − 5 ) 2 1 + 5 = 2 ( x − 5 ) 2 1 = − 3 [ ( x − 5 ) 2 1 ] 2 = ( − 3 ) 2
Our goal is to find a possible solution of the equation based on the work shown.
Simplifying the Equation First, let's simplify the equation [( x − 5 ) 2 1 ] 2 = ( − 3 ) 2 . When we square both sides, we get:
x − 5 = 9
Isolating x Next, we isolate x by adding 5 to both sides of the equation:
x = 9 + 5
x = 14
Checking the Solution Now, let's check if x = 14 is a valid solution by substituting it back into the original equation:
( x − 5 ) 2 1 + 5 = 2
( 14 − 5 ) 2 1 + 5 = ( 9 ) 2 1 + 5 = 3 + 5 = 8
Since 8 = 2 , x = 14 is not a valid solution to the original equation. However, the question asks for a possible solution based on the work shown, so we proceed with x = 14 as a possible solution, even though it's an extraneous solution.
Final Answer Therefore, based on the work shown, a possible solution of the equation is x = 14 .
Examples
When solving equations, it's important to check your solutions to make sure they are valid. Sometimes, you might find a solution that works in the transformed equation but not in the original equation. These are called extraneous solutions. For example, when dealing with square roots or rational expressions, checking solutions is crucial to avoid extraneous solutions. This concept is used in various fields like physics and engineering where equations model real-world phenomena, and incorrect solutions can lead to flawed predictions or designs.
The steps lead to the potential solution x = 14 , which is derived from isolating and squaring the equation. However, this value does not satisfy the original equation, indicating it is an extraneous solution. Therefore, based on the work shown, a possible solution is 14 .
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