OG详解-OG10 数学2 Q24

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.