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

$\frac{x^{2}}{\sqrt{x^{2} - c^{2}}} = \frac{c^{2}}{\sqrt{x^{2} - c^{2}}} + 39$

In the given equation, $c$ is a positive constant. Which of the following is one of the solutions to the given equation?

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Explanation

Choice D is correct. If $x^{2} - c^{2} \leq 0$ , then neither side of the given equation is defined and there can be no solution. Therefore, $x squared minus c squared greater than 0$ . Subtracting $\frac{c^{2}}{\sqrt{x^{2} - c^{2}}}$ from both sides of the given equation yields $\frac{x^{2}}{\sqrt{x^{2} - c^{2}}} - \frac{c^{2}}{\sqrt{x^{2} - c^{2}}} = 39$ , or $\frac{x^{2} - c^{2}}{\sqrt{x^{2} - c^{2}}} = 39$ . Squaring both sides of this equation yields ${(\frac{x^{2} - c^{2}}{\sqrt{x^{2} - c^{2}}})}^{2} = 39^{2}$ , or $\frac{(x^{2} - c^{2}) (x^{2} - c^{2})}{x^{2} - c^{2}} = 39^{2}$ . Since $x^{2} - c^{2}$ is positive and, therefore, nonzero, the expression $\frac{x^{2} - c^{2}}{x^{2} - c^{2}}$ is defined and equivalent to $1$ . It follows that the equation $\frac{(x^{2} - c^{2}) (x^{2} - c^{2})}{x^{2} - c^{2}} = 39^{2}$ can be rewritten as $(\frac{x^{2} - c^{2}}{x^{2} - c^{2}}) (x^{2} - c^{2}) = 39^{2}$ , or $(1) (x^{2} - c^{2}) = 39^{2}$ , which is equivalent to $x^{2} - c^{2} = 39^{2}$ . Adding $c^{2}$ to both sides of this equation yields $x^{2} = c^{2} + 39^{2}$ . Taking the square root of both sides of this equation yields two solutions: $x = \sqrt{c^{2} + 39^{2}}$ and $x = - \sqrt{c^{2} + 39^{2}}$ . Therefore, of the given choices, $- \sqrt{c^{2} + 39^{2}}$ is one of the solutions to the given equation.

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

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

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