- 9 x 2 + 30 x + c = 0 In the given equation, c is a constant. The

Question

- 9 x 2 + 30 x + c = 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?

Accepted Answer

Choice C is correct. It's given that the equation - 9 x 2 + 30 x + c = 0 has exactly one solution. A quadratic equation of the form a x 2 + b x + c = 0 has exactly one solution if and only if its discriminant, - 4 a c + b 2 , 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 b 2 - 4 a c yields 30 2 - 4 - 9 c , or 900 + 36 c . Since the discriminant must equal zero, 900 + 36 c = 0 . Subtracting 36 c from both sides of this equation yields 900 = - 36 c . 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.

Question 9-x-2-