document.write( "Question 1209627: Find a complex number $z$ such that the real part and imaginary part of $z$ are both integers, and such that
\n" ); document.write( "z^2 + 2z = -53 + 8i. \n" ); document.write( "
Algebra.Com's Answer #849714 by mccravyedwin(408)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"z%5E2+%2B+2z+=+-53+%2B+8i\"\r\n" );
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document.write( "Let z = p + qi where p and q are integers.\r\n" );
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document.write( "\"%28p%2Bqi%29%5E2+%2B+2%28p%2Bqi%29+=+-53+%2B+8i\"\r\n" );
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document.write( "\"p%5E2%2B2pqi%2Bq%5E2i%5E2%2B2p%2B2qi=-53%2B8i\"\r\n" );
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document.write( "\"p%5E2%2B2pqi%2Bq%5E2%28-1%29%2B2p%2B2qi=-53%2B8i\"\r\n" );
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document.write( "\"p%5E2%2B2pqi-q%5E2%2B2p%2B2qi=-53%2B8i\"\r\n" );
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document.write( "Set the real parts equal on both sides and set\r\n" );
document.write( "the imaginary parts on both sides equal\r\n" );
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document.write( "\"system%28p%5E2-q%5E2%2B2p=-53%2C2pqi%2B2qi=8i%29\"\r\n" );
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document.write( "Divide the second equation through by 2i\r\n" );
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document.write( "\"system%28p%5E2-q%5E2%2B2p=-53%2Cpq%2Bq=4%29\"\r\n" );
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document.write( "Solve the second equation for q\r\n" );
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document.write( "\"pq%2Bq=4\"\r\n" );
document.write( "\"q%28p%2B1%29=4\"\r\n" );
document.write( "\"q=4%2F%28p%2B1%29\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"p%5E2-q%5E2%2B2p=-53\"\r\n" );
document.write( "\"p%5E2-%284%2F%28p%2B1%29%29%5E2%2B2p=-53\"\r\n" );
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document.write( "By technology, there are two solutions, neither one integers\r\n" );
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document.write( "p = -0.446924168, which gives q = 7.232281305\r\n" );
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document.write( "and\r\n" );
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document.write( "p = -1.553075832, which gives q = -7.232281305\r\n" );
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document.write( "So, there are no such integer solutions, unless somebody can \r\n" );
document.write( "find a mistake that I've made.\r\n" );
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document.write( "Edwin
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