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Advanced Math / Nonlinear functions Difficulty: Hard

The product of two positive integers is $462$ . If the first integer is $5$ greater than twice the second integer, what is the smaller of the two integers?

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Explanation

The correct answer is $14$ . Let $x$ represent the first integer and $y$ represent the second integer. If the first integer is $5$ greater than twice the second integer, then $x = 2 y + 5$ . It's given that the product of the two integers is $462$ ; therefore $x y equals 462$ . Substituting $2 y plus 5$ for $x$ in this equation yields $(2 y + 5) (y) = 462$ , which can be written as $2 y^{2} + 5 y = 462$ . Subtracting $462$ from each side of this equation yields $2 y^{2} + 5 y - 462 = 0$ . The left-hand side of this equation can be factored by finding two values whose product is $2 left parenthesis negative 462 right parenthesis$ , or $negative 924$ , and whose sum is $5$ . The two values whose product is $negative 924$ and whose sum is $5$ are $33$ and $negative 28$ . Thus, the equation $2 y^{2} + 5 y - 462 = 0$ can be rewritten as $2 y^{2} - 28 y + 33 y - 462 = 0$ , which is equivalent to $2 y (y - 14) + 33 (y - 14) = 0$ , or $(2 y + 33) (y - 14) = 0$ . By the zero product property, it follows that $2 y + 33 = 0$ or $y - 14 = 0$ . Subtracting $33$ from both sides of the equation $2 y + 33 = 0$ yields $2 y = - 33$ . Dividing both sides of this equation by $2$ yields $y = - \frac{33}{2}$ . Since $y$ is a positive integer, the value of $y$ isn't $- \frac{33}{2}$ . Adding $14$ to both sides of the equation $y - 14 = 0$ yields $y equals 14$ . Substituting $14$ for $y$ in the equation $x y equals 462$ yields $x left parenthesis 14 right parenthesis equals 462$ . Dividing both sides of this equation by $14$ yields $x equals 33$ . Therefore, the two integers are $14$ and $33$ , so the smaller of the two integers is $14$ .