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

$- 2 x^{2} + 20 x + c = 0$

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 B is correct. It's given that the equation $- 2 x^{2} + 20 x + c = 0$ , where $c$ is a constant, 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, $b squared minus 4 a c$ , is equal to zero. It follows that for the given equation, $a = - 2$ and $b equals 20$ . Substituting $negative 2$ for $a$ and $20$ for $b$ in $b squared minus 4 a c$ yields $20^{2} - 4 (- 2) (c)$ , or $400 plus 8 c$ . Since the discriminant must equal zero, it follows that $400 + 8 c = 0$ . Subtracting $400$ from both sides of this equation yields $8 c = - 400$ . Dividing each side of this equation by $8$ yields $c = - 50$ . Therefore, the value of $c$ is $negative 50$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.