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

$x - 29 = (x - a) (x - 29)$

Which of the following are solutions to the given equation, where $a$ is a constant and $a greater than 30$ ?

$a$
$a plus 1$
$29$

I and II only

I and III only

II and III only

I, II, and III

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Explanation

Choice C is correct. Subtracting the expression $left parenthesis x minus 29 right parenthesis$ 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 $left parenthesis x minus 29 right parenthesis$ , 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 $x equals 29$ . Adding $a plus 1$ 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 $a plus 1$ . Thus, only $a plus 1$ 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.