document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102471>Question 1102471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The equation &#119911;^2 = &#8722;5 + 12&#119894; has two (complex) solutions. Note that<BR>\n" );
document.write( "&#119911; = sqrt (&#8722;5 + 12&#119894;)is not a particularly useful way to write a solution, since it does not tell us the real and imaginary parts of the solution. We are actually trying to find solutions in the form &#119911; = &#119909; + &#119894;&#119910;. Show that &#119911; = 2 + 3&#119894; is such a solution and find the second solution. I know how to prove that z= 2 +3i is a solution, but I just don't know how to find the other solution? Please explain! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717165 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> &#119911;^2 = &#8722;5 + 12&#119894;<BR>\n" );
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document.write( "To solve this problem, we use the polar form of a complex number and De Moivre's Theorem<BR>\n" );
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document.write( "The polar form of a complex number(a+bi, where a,b are real numbers) is r(cos theta + i * sin theta) where r > or = to zero, r is the distance formula = square root(a^2+b^2) and tan theta = b/a<BR>\n" );
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document.write( "write &#119911;^2 = &#8722;5 + 12&#119894; in polar form<BR>\n" );
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document.write( "r = square root((-5)^2 + 12^2) = square root(25 + 144) = 13<BR>\n" );
document.write( "tan theta = 12/-5 = -2.4 and theta is -67.38<BR>\n" );
document.write( "Note that 180 - 67.38 is 112.62<BR>\n" );
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document.write( "then z^2 = 13(cos (112.62) + i * sin(112.62))<BR>\n" );
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document.write( "using De Moivre's Theorem, we know that z^2 has two roots<BR>\n" );
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document.write( "let n be a positive number(2 for this problem) and let w(k) be the kth root, then<BR>\n" );
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document.write( "w(k) = r^(1/2)[ cos((112.62 + 2kpi)/2) + ( i * sin((112.62 + 2kpi/2)))<BR>\n" );
document.write( "where k = 0, 1 an pi = 180 degrees<BR>\n" );
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document.write( "w(0) = 13(1/2) * (cos(112.62 / 2) + (i * sin(112.62 / 2))<BR>\n" );
document.write( "w(0) = 2 + 3i<BR>\n" );
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document.write( "w(1) = 13(1/2) * (cos((112.62+360) / 2) + (i * sin((112.62+360) / 2)) <BR>\n" );
document.write( "w(1) = -2 -3i<BR>\n" );
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document.write( "the other solution is -2 -3i<BR>\n" );
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document.write( " </SPAN>\n" );
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