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The product of two positive integers is 546. If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?

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The product of two positive integers is 546. If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?

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Choice B is correct. Let $\\bm{x}$ be the first integer and let $y$ be the second integer. If the first integer is $11$ greater than twice the second integer, then $\\bm{x}=2y+11$. If the product of the two integers is $546$, then $\\bm{xy}=546$. Substituting $2y+11$ for $\\bm{x}$ in this equation results in $(2y+11)y=546$. Distributing the $y$ to both terms in the parentheses results in $2y^{2}+11y=546$. Subtracting $546$ from both sides of this equation results in $2y^{2}+11y-546=0$. The left-hand side of this equation can be factored by finding two values whose product is $2(-546)$, or $-1,092$, and whose sum is $11$. The two values whose product is $-1,092$ and whose sum is $11$ are $39$ and $-28$. Thus, the equation $2y^{2}+11y-546=0$ can be rewritten as $2y^{2}+28y-39y-546=0$, which is equivalent to $2y(y-14)+39(y-14)=0$, or $(2y+39)(y-14)=0$. By the zero product property, it follows that $2y+39=0$ and $y-14=0$. Subtracting $39$ from both sides of the equation $2y+39=0$ yields $2y=-39$. Dividing both sides of this equation by $2$ yields $y=-\\frac{39}{2}$. Since $y$ is a positive integer, the value of $y$ is not $-\\frac{39}{2}$. Adding $14$ to both sides of the equation $y-14=0$ yields $y=14$. Substituting $14$ for $y$ in the equation $\\bm{xy}=546$ yields $x(14)=546$. Dividing both sides of this equation by $14$ results in $\\bm{x}=39$. Therefore, the two integers are $14$ and $39$, so the smaller of the two integers is $14$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the larger of the two integers.

Choice D is incorrect and may result from conceptual or calculation errors.

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The product of two positive integers is 546. If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?

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