SOLUTION: There are 2 consecutive positive odd integers such that the square of the smaller integer is four more than five times the larger. What is the sum of the 2 integers?

Question 546011: There are 2 consecutive positive odd integers such that the square of the smaller integer is four more than five times the larger. What is the sum of the 2 integers?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Consecutive odd integers are two units apart. Example 3 & 5 or 101 & 103.
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That being the case, if n represents the first positive odd integer, then n + 2 represents the next positive integer. n +2 is the larger odd integer.
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The problem tells you that the first of these two integers (that means n) is to be squared. Squaring n results in
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Next you are told that this quantity is to be equal to "4 more than 5 times the larger." Well then, 5 times the larger is 5*(n + 2) and 4 more than that is 4 + 5*(n + 2). This is to equal n squared. So we can write the equation:
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Multiply out the 5*(n + 2) on the right side. It becomes 5n + 10. So this makes the equation:
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Add the 4 and the 10 on the right side to get:
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This is a quadratic relationship. Put it into standard quadratic form by subtracting 5n + 14 from both sides to get:
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The left side of this equation can be factored as follows:
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and this equation will be true whenever one of the factors equals zero. This is because multiplication by a zero on the left side makes the entire left side equal to the zero on the right side. This means there are two possible solutions. Either:
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which means that
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or
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which means that
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But the problem says that n must be positive and odd. The only answer that satisfies both those conditions is .
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So if then the next consecutive odd number is .
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The problem asks for the sum of these two digits so the answer is
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Let's just check to make sure that the two digits satisfy the conditions of the problem.
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and that is
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and that also equals
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Therefore, we can say that we worked it out correctly. We have the two correct positive odd integers, and their sum (which the problem asked us to find) is 16.
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Hope this helps you to understand how to work this problem and gives you some insight into how to solve similar problems.
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