Find all real solutions. (Enter your answers as comma-separated lists. If there is no real solution, enter NO REALSOLUTION.)
[tex]
\begin{array}{l}
2-\sqrt{x-2}=x \\
x=?
\end{array}
[/tex]
Answer (1)
Isolate the square root: x − 2 = 2 − x .
Square both sides: x − 2 = ( 2 − x ) 2 .
Solve the quadratic equation: x 2 − 5 x + 6 = 0 , which factors to ( x − 2 ) ( x − 3 ) = 0 .
Check the solutions in the original equation: only x = 2 is a valid solution, so the final answer is 2 .
Explanation
Analyzing the Problem We are given the equation 2 − x − 2 = x and we need to find all real solutions for x . First, we note that the expression inside the square root must be non-negative, so x − 2 ≥ 0 , which means x ≥ 2 .
Isolating the Square Root To solve the equation, we first isolate the square root term: x − 2 = 2 − x . Since the square root is non-negative, we must have 2 − x ≥ 0 , which means x ≤ 2 . Combining this with the condition x ≥ 2 , we have x = 2 .
Checking the Solution Now, we check if x = 2 is a solution to the original equation: 2 − 2 − 2 = 2 − 0 = 2 − 0 = 2 . Since this equals x , the solution x = 2 is valid.
Squaring Both Sides Alternatively, we can square both sides of the equation x − 2 = 2 − x to get x − 2 = ( 2 − x ) 2 . Expanding the right side gives x − 2 = 4 − 4 x + x 2 . Rearranging the equation into a quadratic equation, we have x 2 − 5 x + 6 = 0 .
Solving the Quadratic Equation We can factor the quadratic equation as ( x − 2 ) ( x − 3 ) = 0 . This gives us two possible solutions: x = 2 and x = 3 .
Checking the Solutions Now we need to check if these solutions satisfy the original equation. For x = 2 , we have 2 − 2 − 2 = 2 − 0 = 2 , which is equal to x . So x = 2 is a valid solution. For x = 3 , we have 2 − 3 − 2 = 2 − 1 = 2 − 1 = 1 , which is not equal to x = 3 . So x = 3 is not a valid solution.
Final Answer Therefore, the only real solution to the equation is x = 2 .
Examples
Consider a scenario where you are designing a bridge and need to calculate the length of a support cable. The equation 2 − x − 2 = x might represent a simplified model of the cable's sag, where x is related to the cable's length. Solving this equation ensures that the cable's length meets specific design requirements, guaranteeing the bridge's stability. Understanding how to solve such equations is crucial for ensuring the structural integrity and safety of the bridge.
