document.write( "Question 166951: x^2-x-1=0
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Algebra.Com's Answer #122991 by jim_thompson5910(35256)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "\"x%5E2-x-1\" Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( "Take half of the \"x\" coefficient \"-1\" to get \"-1%2F2\". In other words, \"%281%2F2%29%28-1%29=-1%2F2\".\r
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\n" ); document.write( "\n" ); document.write( "Now square \"-1%2F2\" to get \"1%2F4\". In other words, \"%28-1%2F2%29%5E2=%28-1%2F2%29%28-1%2F2%29=1%2F4\"\r
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\n" ); document.write( "\n" ); document.write( "\"x%5E2-x%2Bhighlight%281%2F4-1%2F4%29-1\" Now add and subtract \"1%2F4\". Make sure to place this after the \"x\" term. Notice how \"1%2F4-1%2F4=0\". So the expression is not changed.\r
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\n" ); document.write( "\n" ); document.write( "\"%28x%5E2-x%2B1%2F4%29-1%2F4-1\" Group the first three terms.\r
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\n" ); document.write( "\n" ); document.write( "\"%28x-1%2F2%29%5E2-1%2F4-1\" Factor \"x%5E2-x%2B1%2F4\" to get \"%28x-1%2F2%29%5E2\".\r
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\n" ); document.write( "\n" ); document.write( "\"%28x-1%2F2%29%5E2-5%2F4\" Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "So after completing the square, \"x%5E2-x-1\" transforms to \"%28x-1%2F2%29%5E2-5%2F4\". So \"x%5E2-x-1=%28x-1%2F2%29%5E2-5%2F4\".\r
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\n" ); document.write( "\n" ); document.write( "So \"x%5E2-x-1=0\" is equivalent to \"%28x-1%2F2%29%5E2-5%2F4=0\".\r
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\n" ); document.write( "\n" ); document.write( "\"%28x-1%2F2%29%5E2-5%2F4=0\" Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( "\"%28x-1%2F2%29%5E2=0%2B5%2F4\"Add \"5%2F4\" to both sides.\r
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\n" ); document.write( "\n" ); document.write( "\"%28x-1%2F2%29%5E2=5%2F4\" Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "\"x-1%2F2=0%2B-sqrt%285%2F4%29\" Take the square root of both sides.\r
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\n" ); document.write( "\n" ); document.write( "\"x-1%2F2=sqrt%285%2F4%29\" or \"x-1%2F2=-sqrt%285%2F4%29\" Break up the \"plus/minus\" to form two equations.\r
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\n" ); document.write( "\n" ); document.write( "\"x-1%2F2=sqrt%285%29%2F2\" or \"x-1%2F2=-sqrt%285%29%2F2\" Simplify the square root.\r
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\n" ); document.write( "\n" ); document.write( "\"x=1%2F2%2Bsqrt%285%29%2F2\" or \"x=1%2F2-sqrt%285%29%2F2\" Add \"1%2F2\" to both sides.\r
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\n" ); document.write( "\n" ); document.write( "\"x=%281%2Bsqrt%285%29%29%2F%282%29\" or \"x=%281-sqrt%285%29%29%2F%282%29\" Combine the fractions.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So the solutions are \"x=%281%2Bsqrt%285%29%29%2F%282%29\" or \"x=%281-sqrt%285%29%29%2F%282%29\".
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