document.write( "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? \n" ); document.write( "
Algebra.Com's Answer #355842 by bucky(2189)\"\" \"About 
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Consecutive odd integers are two units apart. Example 3 & 5 or 101 & 103.
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\n" ); document.write( "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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\n" ); document.write( "The problem tells you that the first of these two integers (that means n) is to be squared. Squaring n results in \"n%5E2\"
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\n" ); document.write( "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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\n" ); document.write( "\"n%5E2+=+4+%2B+5%2A%28n+%2B+2%29\"
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\n" ); document.write( "Multiply out the 5*(n + 2) on the right side. It becomes 5n + 10. So this makes the equation:
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\n" ); document.write( "\"n%5E2+=+4+%2B+5n+%2B+10\"
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\n" ); document.write( "Add the 4 and the 10 on the right side to get:
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\n" ); document.write( "\"n%5E2+=+5n+%2B+14\"
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\n" ); document.write( "This is a quadratic relationship. Put it into standard quadratic form by subtracting 5n + 14 from both sides to get:
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\n" ); document.write( "\"n%5E2+-+5n+-14+=+0\"
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\n" ); document.write( "The left side of this equation can be factored as follows:
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\n" ); document.write( "\"%28n+-+7%29%2A%28n+%2B+2%29+=+0\"
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\n" ); document.write( "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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\n" ); document.write( "\"n+-+7+=+0\" which means that \"n+=+7\"
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\n" ); document.write( "or
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\n" ); document.write( "\"n+%2B+2+=+0\" which means that \"n+=+-2\"
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\n" ); document.write( "But the problem says that n must be positive and odd. The only answer that satisfies both those conditions is \"n+=+7\".
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\n" ); document.write( "So if \"n+=+7\" then the next consecutive odd number is \"n+=+9\".
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\n" ); document.write( "The problem asks for the sum of these two digits so the answer is \"7%2B9+=+16\"
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\n" ); document.write( "Let's just check to make sure that the two digits satisfy the conditions of the problem.
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\n" ); document.write( "\"n%5E2+=+7%2A7\" and that is \"49\"
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\n" ); document.write( "\"4%2B+5%2A9+=+4+%2B+45\" and that also equals \"49\"
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\n" ); document.write( "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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\n" ); document.write( "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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