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Advanced Math / Equivalent expressions Difficulty: Hard

If $k - x$ is a factor of the expression $- x^{2} + \frac{1}{29} n k^{2}$ , where $n$ and $k$ are constants and $k greater than 0$ , what is the value of $n$ ?

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Explanation

Choice D is correct. If $k - x$ is a factor of the expression $- x^{2} + (\frac{1}{29}) n k^{2}$ , then the expression can be written as $(k - x) (a x + b)$ , where $a$ and $b$ are constants. This expression can be rewritten as $a k x + b k - a x^{2} - b x$ , or $- a x^{2} + (a k - b) x + b k$ . Since this expression is equivalent to $- x^{2} + (\frac{1}{29}) n k^{2}$ , it follows that $- a = - 1$ , $a k - b = 0$ , and $b k = (\frac{1}{29}) n k^{2}$ . Dividing each side of the equation $- a = - 1$ by $negative 1$ yields $a equals 1$ . Substituting $1$ for $a$ in the equation $a k - b = 0$ yields $k - b = 0$ . Adding $b$ to each side of this equation yields $k = b$ . Substituting $k$ for $b$ in the equation $b k = (\frac{1}{29}) n k^{2}$ yields $k^{2} = (\frac{1}{29}) n k^{2}$ . Since $k$ is positive, dividing each side of this equation by $k^{2}$ yields $1 equals left parenthesis one twenty ninth right parenthesis n$ . Multiplying each side of this equation by $29$ yields $29 equals n$ .

Alternate approach: The expression $x^{2} - y^{2}$ can be written as $(x - y) (x + y)$ , which is a difference of two squares. It follows that $(\frac{1}{29}) n k^{2} - x^{2}$ is equivalent to $((\sqrt{\frac{1}{29} n}) k - x) ((\sqrt{\frac{1}{29} n}) k + x)$ . It’s given that $k - x$ is a factor of $- x^{2} + (\frac{1}{29}) n k^{2}$ , so the factor $(\sqrt{\frac{1}{29} n}) k - x$ is equal to $k - x$ . Adding $x$ to both sides of the equation $(\sqrt{\frac{1}{29} n}) k - x = k - x$ yields $(\sqrt{\frac{1}{29} n}) k = k$ . Since $k$ is positive, dividing both sides of this equation by $k$ yields $StartRoot one twenty ninth n EndRoot equals 1$ . Squaring both sides of this equation yields $one twenty ninth n equals 1$ . Multiplying both sides of this equation by $29$ yields $n equals 29$ .

Choice A is incorrect. This value of $n$ gives the expression $- x^{2} + (\frac{1}{29}) (- 29) k^{2}$ , or $- x^{2} - k^{2}$ . This expression doesn't have $k - x$ as a factor.

Choice B is incorrect. This value of $n$ gives the expression $- x^{2} + (\frac{1}{29}) (- \frac{1}{29}) k^{2}$ , or $- x^{2} + (- \frac{1}{841}) k^{2}$ . This expression doesn't have $k - x$ as a factor.

Choice C is incorrect. This value of $n$ gives the expression $- x^{2} + (\frac{1}{29}) (\frac{1}{29}) k^{2}$ , or $- x^{2} + (\frac{1}{841}) k^{2}$ . This expression doesn't have $k - x$ as a factor.