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

Question

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?

Accepted Answer

Choice B is correct. Let 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 x = 2 y + 11 . If the product of the two integers is 546 , then x y = 546 . Substituting 2 y + 11 for x in this equation results in 2 y + 11 y = 546 . Distributing the y to both terms in the parentheses results in 2 y 2 + 11 y = 546 . Subtracting 546 from both sides of this equation results in 2 y 2 + 11 y - 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 2 y 2 + 11 y - 546 = 0 can be rewritten as 2 y 2 + 28 y - 39 y - 546 = 0 , which is equivalent to 2 y y - 14 + 39 y - 14 = 0 , or 2 y + 39 y - 14 = 0 . By the zero product property, it follows that 2 y + 39 = 0 and y - 14 = 0 . Subtracting 39 from both sides of the equation 2 y + 39 = 0 yields 2 y = - 39 . Dividing both sides of this equation by 2 yields y = - 39 2 . Since y is a positive integer, the value of y is not - 39 2 . Adding 14 to both sides of the equation y - 14 = 0 yields y = 14 . Substituting 14 for y in the equation x y = 546 yields x 14 = 546 . Dividing both sides of this equation by 14 results in 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.

Question a8304c