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

$x^{2} - 2 x - 9 = 0$

One solution to the given equation can be written as $1 plus StartRoot k EndRoot$ , where $k$ is a constant. What is the value of $k$ ?

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Explanation

Choice B is correct. Adding $9$ to each side of the given equation yields $x^{2} - 2 x = 9$ . To complete the square, adding $1$ to each side of this equation yields $x^{2} - 2 x + 1 = 9 + 1$ , or ${(x - 1)}^{2} = 10$ . Taking the square root of each side of this equation yields $x - 1 = \pm \sqrt{10}$ . Adding $1$ to each side of this equation yields $x = 1 \pm \sqrt{10}$ . Since it's given that one of the solutions to the equation can be written as $1 plus StartRoot k EndRoot$ , the value of $k$ must be $10$ .

Alternate approach: It's given that $1 plus StartRoot k EndRoot$ is a solution to the given equation. It follows that $x = 1 + \sqrt{k}$ . Substituting $1 plus StartRoot k EndRoot$ for $x$ in the given equation yields ${(1 + \sqrt{k})}^{2} - 2 (1 + \sqrt{k}) - 9 = 0$ , or $(1 + \sqrt{k}) (1 + \sqrt{k}) - 2 (1 + \sqrt{k}) - 9 = 0$ . Expanding the products on the left-hand side of this equation yields $1 + 2 \sqrt{k} + k - 2 - 2 \sqrt{k} - 9 = 0$ , or $k - 10 = 0$ . Adding $10$ to each side of this equation yields $k equals 10$ .

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.