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−9x^2 +30x+c = 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c? 3
Correct answer: C (parsed from the explanation — this source omits an answer key for this item)
Explanation
-9 2+30 + =0 Choice C is correct. It’s given that the equation x x c has exactly 2+ + =0 one solution. A quadratic equation of the form ax bx c has exactly one -4 + 2 solution if and only if its discriminant, ac b , is equal to zero. It follows that for =- =30 - 30 the given equation, a 9 and b . Substituting 9 for a and for b into 2-4 302-4 -9 900+36 b ac yields ^ h^ch, or c. Since the discriminant must equal 900+36 =0 36 zero, c . Subtracting c from both sides of this equation yields 900=-36 -36 -25= c. Dividing each side of this equation by yields c. -25 Therefore, the value of c is . 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 -53 value of c is , this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.
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