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(x^2)/(x^2 −c^2) = (c^2)/(x^2 −c^2) +39 In the given equation, c is a positive constant. Which of the following is one of the solutions to the given equation? −c
Correct answer: D (parsed from the explanation — this source omits an answer key for this item)
Explanation
2- 2#0 Choice D is correct. If x c , then neither side of the given equation is 2- 2>0 c2 defined and there can be no solution. Therefore, x c . Subtracting x2- c2 x 2 - c 2 =39 x 2- c 2 =39 from both sides of the given equation yields 2- 2 2- 2 , or 2- 2 . x c x c x c Squaring both sides of this equation yields e x 2 2 - - c 2 2o 2 =392 , or ^x 2- c 2 2 h - ^x 2 2 - c 2 h=392 . x c x c 2- 2 x2- c2 Since x c is positive and, therefore, nonzero, the expression is defined x2- c2 and equivalent to 1. It follows that the equation ^x 2- c 2 2 h - ^x 2 2 - c 2 h=392 can be rewritten x c x 2- c 2 2- 2 =392 1 2- 2 =392 2- 2=392 as e 2- 2o`x c j , or ^ h^x c h , which is equivalent to x c . x c 2 2= 2+392 Adding c to both sides of this equation yields x c . Taking the square = 2+392 root of both sides of this equation yields two solutions: x c and =- 2+392 - 2+392 x c . Therefore, of the given choices, c is one of the solutions to the given equation. Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.
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