Nonlinear equations in one variable and systems of equations in two variables SAT question

Choice B is correct. Adding $9$ to each side of the given equation yields $x^2-2x=9$. To complete the square, adding $1$ to each side of this equation yields $x^2-2x+1=9+1$, 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 $1+\sqrt{k}$, the value of $k$ must be $10$.

Alternate approach: It's given that $1+\sqrt{k}$ is a solution to the given equation. It follows that $x=1+\sqrt{k}$. Substituting $1+\sqrt{k}$ 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 $k=10$.

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.