document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1081232>Question 1081232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve w^2 = 15+8i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #695265 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=695265\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let w = a+bi be any complex number\r<BR>\n" );
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document.write( "Square both sides<BR>\n" );
document.write( "w = a+bi<BR>\n" );
document.write( "w^2 = (a+bi)^2<BR>\n" );
document.write( "w^2 = (a+bi)(a+bi)<BR>\n" );
document.write( "w^2 = a(a+bi)+bi(a+bi)<BR>\n" );
document.write( "w^2 = a^2 + abi + abi + b^2i^2<BR>\n" );
document.write( "w^2 = a^2 + abi + abi + b^2(-1)<BR>\n" );
document.write( "w^2 = a^2 + abi + abi - b^2<BR>\n" );
document.write( "w^2 = a^2 - b^2 + abi + abi<BR>\n" );
document.write( "w^2 = (a^2 - b^2) + (2ab)i\r<BR>\n" );
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document.write( "Since w^2 = 15+8i, we can say<BR>\n" );
document.write( "(a^2 - b^2) + (2ab)i = 15 + 8i\r<BR>\n" );
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document.write( "The real parts are (a^2-b^2) and 15 for the left and right sides respectively. This must mean they are equal, so,\r<BR>\n" );
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document.write( "a^2 - b^2 = 15\r<BR>\n" );
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document.write( "Likewise, the imaginary parts are (2ab)i and 8i for the left and right sides respectively. So,\r<BR>\n" );
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document.write( "(2ab)i = 8i<BR>\n" );
document.write( "2ab = 8<BR>\n" );
document.write( "2ab/2 = 8/2<BR>\n" );
document.write( "ab = 4\r<BR>\n" );
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document.write( "The system of equations we need to solve is<BR>\n" );
document.write( "a^2 - b^2 = 15<BR>\n" );
document.write( "ab = 4\r<BR>\n" );
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document.write( "Let's solve equation (2) shown above for the variable b<BR>\n" );
document.write( "ab = 4<BR>\n" );
document.write( "ab/a = 4/a<BR>\n" );
document.write( "b = 4/a\r<BR>\n" );
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document.write( "Which is then plugged into the other equation and we solve for 'a'<BR>\n" );
document.write( "a^2 - b^2 = 15<BR>\n" );
document.write( "a^2 - (4/a)^2 = 15<BR>\n" );
document.write( "a^2 - 16/a^2 = 15<BR>\n" );
document.write( "a^2*(a^2 - 16/a^2) = a^2*15<BR>\n" );
document.write( "a^4 - 16 = 15a^2<BR>\n" );
document.write( "a^4 - 16-15a^2 = 15a^2-15a^2<BR>\n" );
document.write( "a^4-15a^2- 16 = 0<BR>\n" );
document.write( "(a^2-16)(a^2+1) = 0<BR>\n" );
document.write( "(a-4)(a+4)(a^2+1) = 0<BR>\n" );
document.write( "a-4 = 0 or a+4 = 0 or a^2+1 = 0<BR>\n" );
document.write( "a = 4 or a = -4 or a = i or a = -i\r<BR>\n" );
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document.write( "If a = 4, then<BR>\n" );
document.write( "b = 4/a<BR>\n" );
document.write( "b = 4/4<BR>\n" );
document.write( "b = 1<BR>\n" );
document.write( "Which means,<BR>\n" );
document.write( "w = a+bi<BR>\n" );
document.write( "w = 4+1i<BR>\n" );
document.write( "w = 4+i<BR>\n" );
document.write( "is one solution\r<BR>\n" );
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document.write( "If a = -4, then<BR>\n" );
document.write( "b = 4/a<BR>\n" );
document.write( "b = 4/(-4)<BR>\n" );
document.write( "b = -1<BR>\n" );
document.write( "Which means,<BR>\n" );
document.write( "w = a+bi<BR>\n" );
document.write( "w = -4-1i<BR>\n" );
document.write( "w = -4-i<BR>\n" );
document.write( "is another solution\r<BR>\n" );
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document.write( "If a = i, then<BR>\n" );
document.write( "b = 4/a<BR>\n" );
document.write( "b = 4/i<BR>\n" );
document.write( "b = (4/i)*(i/i)<BR>\n" );
document.write( "b = (4i)/(i^2)<BR>\n" );
document.write( "b = (4i)/(-1)<BR>\n" );
document.write( "b = -4i<BR>\n" );
document.write( "Which means<BR>\n" );
document.write( "w = a+bi<BR>\n" );
document.write( "w = i+(-4i)i<BR>\n" );
document.write( "w = i-4i^2<BR>\n" );
document.write( "w = i-4(-1)<BR>\n" );
document.write( "w = i+4<BR>\n" );
document.write( "w = 4+i<BR>\n" );
document.write( "is another solution, but as shown earlier, this solution occurs when a = 4\r<BR>\n" );
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document.write( "If a = -i, then<BR>\n" );
document.write( "b = 4/a<BR>\n" );
document.write( "b = 4/i<BR>\n" );
document.write( "b = (4/(-i))*(i/i)<BR>\n" );
document.write( "b = (4i)/(-i^2)<BR>\n" );
document.write( "b = (4i)/(-(-1))<BR>\n" );
document.write( "b = 4i<BR>\n" );
document.write( "Which means<BR>\n" );
document.write( "w = a+bi<BR>\n" );
document.write( "w = -i+(4i)i<BR>\n" );
document.write( "w = -i+4i^2<BR>\n" );
document.write( "w = -i+4(-1)<BR>\n" );
document.write( "w = -i-4<BR>\n" );
document.write( "w = -4-i<BR>\n" );
document.write( "but this is another repeat of what was shown when a = -4\r<BR>\n" );
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document.write( "To summarize, the only two solutions are<BR>\n" );
document.write( "w = 4+i<BR>\n" );
document.write( "w = -4-i\r<BR>\n" );
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document.write( "To condense things a bit, we can write <BR>\n" );
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document.write( "note how -(4+i) = -4-i\r<BR>\n" );
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document.write( " </SPAN>\n" );
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