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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$- 9 x^{2} + 30 x + c = 0$

In the given equation, $c$ is a constant. The equation has exactly one solution. What is the value of $c$ ?

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Explanation

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 $b equals 30$ . Substituting $negative 9$ for $a$ and $30$ for $b$ into $b squared minus 4 a c$ yields $30^{2} - 4 (- 9) (c)$ , or $900 plus 36 c$ . 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 $negative 36$ yields $- 25 = c$ . Therefore, the value of $c$ is $negative 25$ .

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 $negative 53$ , this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.