OG详解-OG4 数学2 Q12

Choice B is correct. Multiplying each side of the given equation by  -16  yields 64x² +112x = 576. To complete the square, adding 49 to each side of this equation yields 64x² +112x + 49 = 576 + 49, or (8x +7)²  = 625. Taking the square root of each side of this equation yields two equations:  8x + 7 = 25  and 8x + 7 =-25. Subtracting 7 from each side of the equation 8x + 7 = 25  yields 8x = 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 8x +7 =-25  yields 8x = -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 is the positive solution to the given equation. Alternate approach: Adding 4x² and 7x to each side of the given equation yields 0 = 4x² +7x -36. The right-hand side of this equation can be rewritten as 4x² +16x -9x -36. Factoring out the common factor of 4x  from the first two terms of this expression and the common factor of -9  from the second two terms yields 4x (x + 4)-9(x + 4). Factoring out the common factor of (x + 4) from these two terms yields the expression (4x -9)(x + 4). Since this expression is equal to 0, it follows that either 4x-9 = 0 or x + 4 = 0. Adding 9 to each side of the equation 4x-9 = 0 yields 4x = 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  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.