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

Choice C is correct. Subtracting the expression $(x-29)$ from both sides of the given equation yields $0=(x-a)(x-29)-(x-29)$, which can be rewritten as $0=(x-a)(x-29)+(-1)(x-29)$. Since the two terms on the right-hand side of this equation have a common factor of $(x-29)$, it can be rewritten as $0=(x-29)(x-a+(-1))$, or $0=(x-29)(x-a-1)$. Since $x-a-1$ is equivalent to $x-(a+1)$, the equation $0=(x-29)(x-a-1)$ can be rewritten as $0=(x-29)(x-(a+1))$. By the zero product property, it follows that $x-29=0$ or $x-(a+1)=0$. Adding $29$ to both sides of the equation $x-29=0$ yields $x=29$. Adding $a+1$ to both sides of the equation $x-(a+1)=0$ yields $x=a+1$. Therefore, the two solutions to the given equation are $29$ and $a+1$. Thus, only $a+1$ and $29$, not $a$, are 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 D is incorrect and may result from conceptual or calculation errors.