document.write( "Question 913204: The sum of two complex number is 4 and their product is 13. What are the two complex numbers?
\n" ); document.write( "(a+bi)(c+di) = 13
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Algebra.Com's Answer #554347 by josgarithmetic(39620)\"\" \"About 
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\"a%2Bbi%2Bc%2Bdi=4\"
\n" ); document.write( "\"a%2Bc%2B%28b%2Bd%29i=4\"
\n" ); document.write( "\"a%2Bc=4\" and \"b%2Bd=0\"
\n" ); document.write( "\"highlight_green%28b=-d%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"%28a%2Bbi%29%28c%2Bdi%29=13\"
\n" ); document.write( "\"ac-bd%2B%28bc%2Bad%29i=13\"
\n" ); document.write( "\"ac-bd=13\" and \"bc%2Bad=0\"\r
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\n" ); document.write( "\n" ); document.write( "Substituting for b into these last two equations
\n" ); document.write( "\"ac-%28-d%29d=13\"
\n" ); document.write( "\"ac%2Bd%5E2=13\"
\n" ); document.write( "and
\n" ); document.write( "\"%28-d%29c%2Bad=0\"
\n" ); document.write( "\"-cd%2Bad=0\"
\n" ); document.write( "and summarizing the THREE equations which use just variables a,c,d is
\n" ); document.write( "\"highlight_green%28system%28a%2Bc=4%2Cac%2Bd%5E2=13%2C-cd%2Bad=0%29%29\"\r
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\n" ); document.write( "\n" ); document.write( "The last equation in the summary of the three-equation system equivalently gives
\n" ); document.write( "\"d%28a-c%29=0\"\r
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\n" ); document.write( "\n" ); document.write( "Looking at the first equation of that system and the last equation and assuming \"highlight_green%28d%3C%3E0%29\", then
\n" ); document.write( "\"system%28a%2Bc=4%2C%28a-c%29d=0%29\"
\n" ); document.write( "with the assumption about d allows to have the system:
\n" ); document.write( "\"system%28a%2Bc=4%2Ca-c=0%29\"\r
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\n" ); document.write( "\n" ); document.write( "Find through adding first of those to the last of those,
\n" ); document.write( "\"2a=4\"
\n" ); document.write( "\"highlight%28a=2%29\"
\n" ); document.write( "and then
\n" ); document.write( "\"a%2Bc=4\"
\n" ); document.write( "\"2%2Bc=4\"
\n" ); document.write( "\"highlight%28c=2%29\"\r
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\n" ); document.write( "\n" ); document.write( "Return to the two equation which came from the product 13.
\n" ); document.write( "-
\n" ); document.write( "\"ac-bd=13\"
\n" ); document.write( "\"2%2A2-bd=13\"
\n" ); document.write( "\"-bd=13-4\"
\n" ); document.write( "\"bd=-9\"----coming back to this after other steps...
\n" ); document.write( "-
\n" ); document.write( "\"-bc%2Bad=0\", also from the product 13 situation
\n" ); document.write( "\"2%2Ab%2B2%2Ad=0\", because we know a and c
\n" ); document.write( "\"2%28b%2Bd%29=0\"
\n" ); document.write( "\"b%2Bd=0\"
\n" ); document.write( "-
\n" ); document.write( "Again \"bd=-9\"
\n" ); document.write( "\"bd=%28-d%29d=-9\"
\n" ); document.write( "\"d%5E2=9\"
\n" ); document.write( "\"highlight%28d=3%29\"----- or negative 3 as well.
\n" ); document.write( "-
\n" ); document.write( "Because we found \"b=-d\"
\n" ); document.write( "\"-b=d\"
\n" ); document.write( "\"-b=3\"
\n" ); document.write( "\"highlight%28b=-3%29\"\r
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\n" ); document.write( "\n" ); document.write( "The variables a, b, c, and d, have now been solved for.
\n" ); document.write( "ANSWER: The two complex numbers are 2-3i and 2+3i.
\n" ); document.write( "\"b=-d\"----which we can soon use in substitution.\r
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