Which of the following is an extraneous solution of $(45-3 x)^{\frac{1}{2}}=x-9$?
A. $x=-12$
B. $x=-3$
C. $x=3$
D. $x=12$
Answer (2)
Square both sides of the equation ( 45 − 3 x ) 2 1 = x − 9 to get 45 − 3 x = ( x − 9 ) 2 .
Simplify the equation to x 2 − 15 x + 36 = 0 .
Solve the quadratic equation to find x = 3 and x = 12 .
Check the solutions in the original equation; x = 3 is extraneous, while x = 12 is valid. The extraneous solution is 3 .
Explanation
Understanding the Problem We are given the equation ( 45 − 3 x ) 2 1 = x − 9 and asked to find the extraneous solution from the options x = − 12 , x = − 3 , x = 3 , and x = 12 . An extraneous solution is a solution that arises when solving an equation but does not satisfy the original equation.
Squaring Both Sides First, let's square both sides of the equation to eliminate the square root: ( 45 − 3 x ) 2 1 = x − 9
( 45 − 3 x ) = ( x − 9 ) 2
Expanding and Simplifying Next, expand and simplify the equation: 45 − 3 x = x 2 − 18 x + 81 0 = x 2 − 15 x + 36
Solving the Quadratic Equation Now, we solve the quadratic equation x 2 − 15 x + 36 = 0 . We can factor this equation as follows: ( x − 3 ) ( x − 12 ) = 0 So, the solutions are x = 3 and x = 12 .
Checking for Extraneous Solutions We need to check if these solutions are extraneous by substituting them back into the original equation ( 45 − 3 x ) 2 1 = x − 9 .
For x = 3 :
( 45 − 3 ( 3 ) ) 2 1 = 3 − 9 ( 45 − 9 ) 2 1 = − 6 ( 36 ) 2 1 = − 6 6 = − 6 This is not true, so x = 3 is an extraneous solution.
For x = 12 :
( 45 − 3 ( 12 ) ) 2 1 = 12 − 9 ( 45 − 36 ) 2 1 = 3 ( 9 ) 2 1 = 3 3 = 3 This is true, so x = 12 is a valid solution.
Identifying the Extraneous Solution Therefore, the extraneous solution is x = 3 .
Examples
When solving equations involving radicals, it's crucial to check for extraneous solutions. For example, consider a scenario where you're designing a rectangular garden. The area is given by A = lw , where l is the length and w is the width. Suppose you have a constraint that relates the length and width through a square root, such as l = w − 2 . After substituting and solving for w , you might find two possible values. However, only one value might make sense in the context of the garden's dimensions (e.g., a negative width is not physically possible). The other value would be an extraneous solution, mathematically correct but not applicable to the real-world scenario.
The extraneous solution from the equation ( 45 − 3 x ) 2 1 = x − 9 is x = 3 . After checking both solutions, x = 3 does not satisfy the original equation, while x = 12 does. Therefore, the correct answer is option C: x = 3 .
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