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)
The solution process leads to x = 14 based on isolating the square root and solving, but upon substituting back into the original equation, it does not satisfy it. Therefore, while 14 is derived as a potential solution, it is not valid. The final answer remains 14 .
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Isolate the square root term: ( x − 5 ) 2 1 = − 3 .
Square both sides: x − 5 = 9 .
Solve for x : x = 14 .
A possible solution of the equation is 14 .
Explanation
Solving for x We are given the equation ( x − 5 ) 2 1 + 5 = 2 and the steps to solve it. Let's analyze the steps and complete the solution.
The first step is to isolate the square root term: ( x − 5 ) 2 1 + 5 = 2 becomes ( x − 5 ) 2 1 = 2 − 5 , which simplifies to ( x − 5 ) 2 1 = − 3 .
The next step is to square both sides of the equation to eliminate the square root: [ ( x − 5 ) 2 1 ] 2 = ( − 3 ) 2 , which simplifies to x − 5 = 9 .
Now, we solve for x :
x − 5 = 9 becomes x = 9 + 5 , so x = 14 .
Now, we need to check if x = 14 is a valid solution to the original equation. Substitute x = 14 into the original equation: ( 14 − 5 ) 2 1 + 5 = 2 ( 9 ) 2 1 + 5 = 2 3 + 5 = 2 8 = 2 Since 8 = 2 , x = 14 is not a solution to the original equation. However, the question asks for a possible solution based on the work shown. The work shown leads to x = 14 .
Finding a Possible Solution The steps shown lead to the equation x − 5 = 9 . Solving for x , we get: x = 9 + 5 x = 14
So, a possible solution of the equation based on the work shown is x = 14 . However, we must remember to check this solution in the original equation to ensure it is valid.
Checking the Solution Let's verify if x = 14 is a solution to the original equation: ( x − 5 ) 2 1 + 5 = 2 ( 14 − 5 ) 2 1 + 5 = 2 9 + 5 = 2 3 + 5 = 2 8 = 2
Since 8 = 2 , x = 14 is not a solution to the original equation. However, the question asks for a possible solution based on the steps shown, which leads to x = 14 .
Final Answer Therefore, based on the work shown, a possible solution of the equation is 14 .
Examples
When solving equations, it's crucial to check your solutions to make sure they are valid. Sometimes, when we perform operations like squaring both sides of an equation, we can introduce extraneous solutions that don't actually satisfy the original equation. For example, if we have an equation like x = − 2 , squaring both sides gives x = 4 . However, substituting x = 4 back into the original equation gives 4 = 2 , which is not equal to − 2 . Therefore, x = 4 is an extraneous solution. This concept is used in various fields, such as physics and engineering, where equations often arise that need to be solved and verified to ensure the solutions are physically meaningful.
