OG详解-OG3 数学2 Q26

Choice C is correct. It’s given that the equation -9x² +30x+c = 0 has exactly one solution. A quadratic equation of the form ax² + bx +c = 0 has exactly one  solution if and only if its discriminant,  -4ac + b² , is equal to zero. It follows that for the given equation, a =-9 and b = 30. Substituting -9  for a and 30 for b into b² -4ac  yields  30² -4(-9)(c), or 900 +36c. Since the discriminant must equal zero, 900 +36c = 0. Subtracting 36c from both sides of this equation yields 900 =-36c. Dividing each side of this equation by -36 yields -25 = c. Therefore, the value of c is -25.

Choice A is incorrect. If the value of c is 3, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather  than exactly one solution. Choice B is incorrect. If the value of c is  0, this would  yield a discriminant that is greater than zero. Therefore, the given equation would  have two solutions, rather than exactly one solution. Choice D is incorrect. If the value of c is -53, this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.