- 4 x 2 - 7 x = - 36 What is the positive solution to the given equation?

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Accepted Answer

Choice B is correct. Multiplying each side of the given equation by - 16 yields 64 x 2 + 112 x = 576 . To complete the square, adding 49 to each side of this equation yields 64 x 2 + 112 x + 49 = 576 + 49 , or 8 x + 7 2 = 625 . Taking the square root of each side of this equation yields two equations: 8 x + 7 = 25 and 8 x + 7 = - 25 . Subtracting 7 from each side of the equation 8 x + 7 = 25 yields 8 x = 18 . Dividing each side of this equation by 8 yields x = 18 8 , or x = 9 4 . Therefore, 9 4 is a solution to the given equation. Subtracting 7 from each side of the equation 8 x + 7 = - 25 yields 8 x = - 32 . Dividing each side of this equation by 8 yields x = - 4 . Therefore, the given equation has two solutions, 9 4 and - 4 . Since 9 4 is positive, it follows that 9 4 is the positive solution to the given equation. Alternate approach: Adding 4 x 2 and 7 x to each side of the given equation yields 0 = 4 x 2 + 7 x - 36 . The right-hand side of this equation can be rewritten as 4 x 2 + 16 x - 9 x - 36 . Factoring out the common factor of 4 x from the first two terms of this expression and the common factor of - 9 from the second two terms yields 4 x x + 4 - 9 x + 4 . Factoring out the common factor of x + 4 from these two terms yields the expression 4 x - 9 x + 4 . Since this expression is equal to 0 , it follows that either 4 x - 9 = 0 or x + 4 = 0 . Adding 9 to each side of the equation 4 x - 9 = 0 yields 4 x = 9 . Dividing each side of this equation by 4 yields x = 9 4 . Therefore, 9 4 is a positive solution to the given equation. Subtracting 4 from each side of the equation x + 4 = 0 yields x = - 4 . Therefore, the given equation has two solutions, 9 4 and - 4 . Since 9 4 is positive, it follows that 9 4 is the positive solution to the given equation. Choice A is incorrect. Substituting 7 4 for x in the given equation yields - 49 2 = - 36 , which is false. Choice C is incorrect. Substituting 4 for x in the given equation yields - 92 = - 36 , which is false. Choice D is incorrect. Substituting 7 for x in the given equation yields - 245 = - 36 , which is false.

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