document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209626>Question 1209626</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all complex numbers z satisfying the equation<BR>\n" );
document.write( "\frac{z + 1}{z - 1} = 2 + i + \frac{-7 + 3z}{z}.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849734 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve the equation:\r<BR>\n" );
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document.write( "1. **Multiply both sides to eliminate the fractions:**\r<BR>\n" );
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document.write( "   Multiply both sides of the equation by z(z - 1) to clear the denominators.  Assume z ≠ 0 and z ≠ 1.\r<BR>\n" );
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document.write( "   z(z + 1) = (2 + i)z(z - 1) + (-7 + 3z)(z - 1)\r<BR>\n" );
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document.write( "2. **Expand the terms:**\r<BR>\n" );
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document.write( "   z² + z = (2 + i)(z² - z) + (-7z + 7 + 3z² - 3z)<BR>\n" );
document.write( "   z² + z = 2z² - 2z + iz² - iz + (-10z + 7 + 3z²)<BR>\n" );
document.write( "   z² + z = 5z² - 12z + 7 + iz² - iz\r<BR>\n" );
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document.write( "3. **Rearrange the equation:**\r<BR>\n" );
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document.write( "   Move all terms to one side to set the equation equal to zero:\r<BR>\n" );
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document.write( "   0 = 4z² - 13z + 7 + iz² - iz<BR>\n" );
document.write( "   0 = (4 + i)z² - (13 + i)z + 7\r<BR>\n" );
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document.write( "4. **Use the quadratic formula:**\r<BR>\n" );
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document.write( "   The quadratic formula for complex numbers is the same as for real numbers:\r<BR>\n" );
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document.write( "   z = (-b ± √(b² - 4ac)) / 2a\r<BR>\n" );
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document.write( "   In our case:<BR>\n" );
document.write( "   * a = (4 + i)<BR>\n" );
document.write( "   * b = -(13 + i)<BR>\n" );
document.write( "   * c = 7\r<BR>\n" );
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document.write( "   z = (13 + i ± √((13 + i)² - 4(4 + i)(7))) / 2(4 + i)\r<BR>\n" );
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document.write( "5. **Simplify the expression:**\r<BR>\n" );
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document.write( "   z = (13 + i ± √(169 + 26i - 1 - 112 - 28i)) / (8 + 2i)<BR>\n" );
document.write( "   z = (13 + i ± √(56 - 2i)) / (8 + 2i)\r<BR>\n" );
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document.write( "6. **Simplify the square root (this is the tricky part):**\r<BR>\n" );
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document.write( "   Let √(56 - 2i) = x + yi, where x and y are real numbers.<BR>\n" );
document.write( "   Then (x + yi)² = 56 - 2i<BR>\n" );
document.write( "   x² + 2xyi - y² = 56 - 2i<BR>\n" );
document.write( "   So, x² - y² = 56 and 2xy = -2, which means xy = -1 and y = -1/x.\r<BR>\n" );
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document.write( "   Substitute y = -1/x into x² - y² = 56:<BR>\n" );
document.write( "   x² - 1/x² = 56<BR>\n" );
document.write( "   x⁴ - 56x² - 1 = 0\r<BR>\n" );
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document.write( "   Let u = x². Then u² - 56u - 1 = 0<BR>\n" );
document.write( "   u = (56 ± √(56² + 4))/2 = (56 ± √3136+4)/2 = (56 ± √3140)/2 = 28 ± √785\r<BR>\n" );
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document.write( "   Since x is real, x² must be positive, so we take the positive solution.<BR>\n" );
document.write( "   x = √(28 + √785). Then y = -1/x = -1/√(28 + √785)\r<BR>\n" );
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document.write( "   So, √(56 - 2i) = √(28 + √785) - i/√(28 + √785).\r<BR>\n" );
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document.write( "7. **Substitute back and solve for z:**\r<BR>\n" );
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document.write( "   Substitute the simplified square root back into the expression for z and simplify. This will give you two solutions for z.  The algebra will be a bit messy, but it's straightforward.  Remember to multiply the numerator and denominator by the conjugate of the denominator to simplify the final results.\r<BR>\n" );
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document.write( "Because the calculations are somewhat involved, it is recommended to use a calculator or software to complete the calculations and obtain the numerical values for the complex solutions.<BR>\n" );
document.write( " </SPAN>\n" );
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