document.write( "Question 78113: Completing the square\r
\n" ); document.write( "\n" ); document.write( "2x² - 8x - 9 = 0
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Algebra.Com's Answer #56020 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
\"2x%5E2+-+8x+-+9+=+0\"
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\n" ); document.write( "Don't forget, when you do a problem using the method of completing the square, you want the
\n" ); document.write( "coefficient or multiplier of the \"x%5E2\" term to be 1. Therefore, in this problem
\n" ); document.write( "the first thing you want to do is to divide both sides of the equation by 2. When you do
\n" ); document.write( "that the problem becomes:
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\n" ); document.write( "\"x%5E2+-+4x+-%289%2F2%29+=+0\"
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\n" ); document.write( "Next, look at the x term. Its multiplier or coefficient is -4. Divide it by 2 and square
\n" ); document.write( "the result. The division by 2 gives you -2, and squaring the -2 results in +4.
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\n" ); document.write( "What you do next is to add +4 to the left side. But, if you add +4 then you need to
\n" ); document.write( "also subtract 4 so you don't change the equation. When you do this you have:
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\n" ); document.write( "\"x%5E2+-4x+%2B+4+-+4+-%289%2F2%29+=+0\"
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\n" ); document.write( "For the time being you don't want to mess with the \"x%5E2+-4x+%2B+4\" because that is
\n" ); document.write( "your perfect square. But you can combine the \"-4+-+%289%2F2%29\" to get:
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\n" ); document.write( "\"-4+-+%289%2F2%29+=+-8%2F2+-+9%2F2+=+-17%2F2\"
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\n" ); document.write( "Substituting this into the equation results in:
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\n" ); document.write( "\"x%5E2+-4x+%2B+4+-%2817%2F2%29+=+0\"
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\n" ); document.write( "You want to get it so only the perfect square is on the left side. So eliminate the term
\n" ); document.write( "\"-17%2F2\" from the left side by adding \"%2817%2F2%29\" to both sides to get:
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\n" ); document.write( "\"x%5E2+-+4x+%2B+4+=+%2817%2F2%29\"
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\n" ); document.write( "Now convert the left side to its equivalent square and this results in:
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\n" ); document.write( "\"%28x-2%29%5E2+=+17%2F2\"
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\n" ); document.write( "Note that the -2 is the same as we got by taking half of the coefficient of the -4x in
\n" ); document.write( "the original problem when we were working on creating the perfect square.
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\n" ); document.write( "Now you can solve this for x by taking the square root of both sides. When you do that
\n" ); document.write( "you get:
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\n" ); document.write( "\"x+-+2+=+sqrt%2817%2F2%29\" and \"x+-+2+=+-sqrt%2817%2F2%29\"
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\n" ); document.write( "Notice that the two results show that you are going to get two answers that differ
\n" ); document.write( "only by the sign on the square root term ... one with a + and one with a - sign.
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\n" ); document.write( "Finally, to solve for just x, add +2 to both sides to get:
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\n" ); document.write( "\"x+=+2%2B-sqrt%2817%2F2%29\"
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\n" ); document.write( "It is standard practice not to have a radical term in the denominator. So multiply
\n" ); document.write( "the \"sqrt%2817%2F2%29\" by \"sqrt%282%29%2Fsqrt%282%29\" and it becomes:
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\n" ); document.write( "Substituting this results into the equation for x results in the more standard form of:
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\n" ); document.write( "\"x+=+2+%2B-+sqrt%2834%29%2F2\"
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\n" ); document.write( "and putting all terms over the common denominator of 2 gives you:
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\n" ); document.write( "\"x+=+%284+%2B-sqrt%2834%29%29%2F2\"
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\n" ); document.write( "and that is the form you are more likely to get by using the quadratic formula ... a
\n" ); document.write( "formula that you will likely learn soon if you haven't already.
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\n" ); document.write( "Hope these details help you to understand the process of completing the square a little more.
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