(c) [tex]$3-\sqrt{x-3}=x$[/tex]
[tex]$x=$[/tex]
Answer (1)
Isolate the square root: x − 3 = 3 − x .
Determine the domain: x ≥ 3 and x ≤ 3 , thus x = 3 .
Verify the solution: 3 − 3 − 3 = 3 , which is true.
The solution is 3 .
Explanation
Domain of the equation We are given the equation 3 − x − 3 = x and we need to find the value(s) of x that satisfy this equation. First, we need to consider the domain of the square root function. The expression inside the square root must be non-negative, so x − 3 ≥ 0 , which means x ≥ 3 .
Isolating the square root Now, let's isolate the square root term: x − 3 = 3 − x Since the square root is non-negative, we must have 3 − x ≥ 0 , which means x ≤ 3 . Combining this with the domain x ≥ 3 , we find that the only possible solution is x = 3 .
Checking the solution Let's check if x = 3 is indeed a solution by substituting it into the original equation: 3 − 3 − 3 = 3 − 0 = 3 − 0 = 3 Since this equals x , the value x = 3 is a solution.
Solving the quadratic equation Alternatively, we can square both sides of the equation x − 3 = 3 − x to get rid of the square root: ( x − 3 ) 2 = ( 3 − x ) 2 x − 3 = 9 − 6 x + x 2 Rearranging the equation into a quadratic equation, we have: x 2 − 7 x + 12 = 0 Factoring the quadratic equation, we get: ( x − 3 ) ( x − 4 ) = 0 So the possible solutions are x = 3 or x = 4 .
Verifying the solutions However, we must check these solutions in the original equation. We already verified that x = 3 is a solution. Now let's check x = 4 : 3 − 4 − 3 = 3 − 1 = 3 − 1 = 2 Since 2 = 4 , x = 4 is not a solution. Therefore, the only solution is x = 3 .
Final Answer Thus, the solution to the equation 3 − x − 3 = x is x = 3 .
Examples
Understanding how to solve equations with square roots is crucial in many areas of physics and engineering. For example, when calculating the velocity of an object falling under gravity with air resistance, you often encounter equations involving square roots. Similarly, in electrical engineering, analyzing circuits with inductors and capacitors can lead to equations that require isolating and solving for variables within square roots. Mastering these algebraic techniques allows engineers and physicists to accurately model and predict the behavior of real-world systems.
