document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=397698>Question 397698</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do i solve the equation _ What is the square root of 4-7i? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #281790 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=281790\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is indeed possible to solve it using Euler's formula, by multiplying/dividing by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2865%29 ALIGN=MIDDLE  ALT=\"sqrt%2865%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to obtain a unit vector and obtaining:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+-+7i+=+sqrt%2865%29%284%2Fsqrt%2865%29+-+%287i%2Fsqrt%2865%29%29i%29+=+sqrt%2865%29e%5Ei%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+%28theta%29+=+4%2Fsqrt%2865%29 ALIGN=MIDDLE  ALT=\"cos+%28theta%29+=+4%2Fsqrt%2865%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin+%28theta%29+=+-7%2Fsqrt%2865%29 ALIGN=MIDDLE  ALT=\"sin+%28theta%29+=+-7%2Fsqrt%2865%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "However, the easiest way is probably to assume that the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+-+7i ALIGN=MIDDLE  ALT=\"4+-+7i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+bi ALIGN=MIDDLE  ALT=\"a+%2B+bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > for real numbers a,b. Squaring both sides,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%5E2+=+4+-+7i ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%5E2+=+4+-+7i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B2abi+-+b%5E2+=+4+-+7i ALIGN=MIDDLE  ALT=\"a%5E2+%2B2abi+-+b%5E2+=+4+-+7i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Equating the real and imaginary parts, we get the system of equations\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+b%5E2+=+4 ALIGN=MIDDLE  ALT=\"a%5E2+-+b%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ab+=+-7 ALIGN=MIDDLE  ALT=\"2ab+=+-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> ab = -7/2.\r<BR>\n" );
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document.write( "Using the substitution b = -7/2a,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+%28-7%2F2a%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"a%5E2+-+%28-7%2F2a%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+49%2F4a%5E2+-+4+=+0 ALIGN=MIDDLE  ALT=\"a%5E2+-+49%2F4a%5E2+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E4+-+16a%5E2+-+49+=+0 ALIGN=MIDDLE  ALT=\"4a%5E4+-+16a%5E2+-+49+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then take the square root. Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be positive based on our assumption that a and b are real numbers. After that, the rest is just arithmetic. </SPAN>\n" );
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