document.write( "Question 1043213: \"%28%283%2Bi%29%2F%282-i%29%29%28a%2Bbi%29=1\" In the equation above, a and b are constants. If \"i=sqrt%28-1%29\". what is the value of a? \n" ); document.write( "
Algebra.Com's Answer #658323 by ikleyn(52879)\"\" \"About 
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\n" ); document.write( "\"%28%283%2Bi%29%2F%282-i%29%29%28a%2Bbi%29=1\" In the equation above, a and b are constants. If \"i=sqrt%28-1%29\". what is the value of a?
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document.write( "If  \"%28%283%2Bi%29%2F%282-i%29%29%28a%2Bbi%29\" = \"1\"  then  (a+bi) is the number reciprocal to  \"%28%283%2Bi%29%2F%282-i%29%29\":\r\n" );
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document.write( "a + bi = \"1%2F%28%28%283%2Bi%29%2F%282-i%29%29%29\"  =  \"%282-i%29%2F%283%2Bi%29\",\r\n" );
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document.write( "and all we need to do is to rid of the denominator in  the fraction  \"%282-i%29%2F%283%2Bi%29\".\r\n" );
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document.write( "For it, multiply the fraction by  \"%283-i%29%2F%283-i%29\".  Since the last fraction is 1, multiplication will not change our number  \"%282-i%29%2F%283%2Bi%29\".\r\n" );
document.write( "People also say \"let us multiply the numerator and the denominator of an original fraction by (3-i) \").\r\n" );
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document.write( "So, we have  \r\n" );
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document.write( "   \"%282-i%29%2F%283%2Bi%29\" = \"%28%282-i%29%2A%283-i%29%29%2F%28%283%2Bi%29%2A%283-i%29%29\" = \"%282%2A3+-2i+-+3i+%2B+i%5E2%29%2F%283%5E2+-+i%5E2%29\"\r\n" );
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document.write( "and since \"i%5E2\" = -1,\r\n" );
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document.write( "= \"%286+-+5i+-+1%29%2F%289%2B1%29\" = \"%285-5i%29%2F10%29\" = 0.5 - 0.5*i.\r\n" );
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document.write( "Thus we got  a + bi = 0.5 - 0.5*i.\r\n" );
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document.write( "It implies  a = 0.5,  b = -0.5.  The problem is solved.  Your answer is: a = 0.5,  b = -0.5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "On complex numbers, you have these lessons\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetic operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "in this site.
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