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Question
What is the solution set for the above equation?
A) {5} B) {20} C) {−5, 20} D) {5, 20} Answer
Squaring each side of
\[x  12 = \sqrt{x + 44}\]
\[(x  12)^2 = (\sqrt{x + 44})^2 = x + 44\]
\[(x^2  24x + 144 = x + 44\]
\[(x^2  25x + 100 = 0\]
\[(x  5)( x  20) = 0\]
The solutions to the quadratic are x = 5 and x = 20. However, since the first step was to square each side of the given equation, which is not a reversible operation, you need to check x = 5 and x = 20 in the original equation. Substituting 5 for x gives
\[5  12 = \sqrt{( 5 + 44)} \]
\[7 = \sqrt{49} \]
This is not a true statement (since √49 represents only the positive square root, 7), so x =5 is not a real solution to
\[x  12 = \sqrt{x + 44}\]
Substituting 20 for x gives:
\[20  12 = \sqrt{20 + 44}\]
\[8 = \sqrt{64}\]
This is a true statement, so x = 20 is a solution to x − 12 = √(x + 44). Therefore, the solution set is {20}, which is choice B.
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