document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209770>Question 1209770</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all (real or nonreal) x satisfying<BR>\n" );
document.write( "(x - 3)^4 + (x - 5)^4 = -8 + 6(x - 3)(x - 5)^3 - 11(x - 3)^3 (x - 5). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850267 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=850267\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this equation step-by-step:\r<BR>\n" );
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document.write( "**1. Simplify the Equation:**\r<BR>\n" );
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document.write( "Let a = x - 4. Then:\r<BR>\n" );
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document.write( "* x - 3 = a + 1<BR>\n" );
document.write( "* x - 5 = a - 1\r<BR>\n" );
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document.write( "Substitute these into the equation:\r<BR>\n" );
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document.write( "(a + 1)^4 + (a - 1)^4 = -8 + 6(a + 1)(a - 1)^3 - 11(a + 1)^3 (a - 1)\r<BR>\n" );
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document.write( "Expand the terms:\r<BR>\n" );
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document.write( "* (a + 1)^4 = a^4 + 4a^3 + 6a^2 + 4a + 1<BR>\n" );
document.write( "* (a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1<BR>\n" );
document.write( "* (a + 1)^4 + (a - 1)^4 = 2a^4 + 12a^2 + 2\r<BR>\n" );
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document.write( "* (a - 1)^3 = a^3 - 3a^2 + 3a - 1<BR>\n" );
document.write( "* (a + 1)^3 = a^3 + 3a^2 + 3a + 1\r<BR>\n" );
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document.write( "Substitute these into the right side:\r<BR>\n" );
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document.write( "-8 + 6(a + 1)(a^3 - 3a^2 + 3a - 1) - 11(a^3 + 3a^2 + 3a + 1)(a - 1)<BR>\n" );
document.write( "-8 + 6(a^4 - 3a^3 + 3a^2 - a + a^3 - 3a^2 + 3a - 1) - 11(a^4 + 3a^3 + 3a^2 + a - a^3 - 3a^2 - 3a - 1)<BR>\n" );
document.write( "-8 + 6(a^4 - 2a^3 + 2a - 1) - 11(a^4 + 2a^3 - 2a - 1)<BR>\n" );
document.write( "-8 + 6a^4 - 12a^3 + 12a - 6 - 11a^4 - 22a^3 + 22a + 11<BR>\n" );
document.write( "-5a^4 - 34a^3 + 34a - 3\r<BR>\n" );
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document.write( "Now, the equation becomes:\r<BR>\n" );
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document.write( "2a^4 + 12a^2 + 2 = -5a^4 - 34a^3 + 34a - 3<BR>\n" );
document.write( "7a^4 + 34a^3 + 12a^2 - 34a + 5 = 0\r<BR>\n" );
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document.write( "**2. Factor the Equation:**\r<BR>\n" );
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document.write( "Notice that the coefficients are symmetric. This suggests that we can factor by grouping.\r<BR>\n" );
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document.write( "Let's divide by a^2:\r<BR>\n" );
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document.write( "7a^2 + 34a + 12 - 34/a + 5/a^2 = 0<BR>\n" );
document.write( "7(a^2 + 5/7a^2) + 34(a - 1/a) + 12 = 0\r<BR>\n" );
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document.write( "Let b = a - 1/a. Then b^2 = a^2 - 2 + 1/a^2, so a^2 + 1/a^2 = b^2 + 2\r<BR>\n" );
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document.write( "7(b^2 + 2) + 34b + 12 = 0<BR>\n" );
document.write( "7b^2 + 14 + 34b + 12 = 0<BR>\n" );
document.write( "7b^2 + 34b + 26 = 0\r<BR>\n" );
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document.write( "Using the quadratic formula, we have:\r<BR>\n" );
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document.write( "b = (-34 ± √(34^2 - 4*7*26)) / (2*7)<BR>\n" );
document.write( "b = (-34 ± √(1156 - 728)) / 14<BR>\n" );
document.write( "b = (-34 ± √428) / 14<BR>\n" );
document.write( "b = (-34 ± 2√107) / 14<BR>\n" );
document.write( "b = (-17 ± √107) / 7\r<BR>\n" );
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document.write( "**3. Solve for a:**\r<BR>\n" );
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document.write( "* a - 1/a = b<BR>\n" );
document.write( "* a^2 - 1 = ba<BR>\n" );
document.write( "* a^2 - ba - 1 = 0\r<BR>\n" );
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document.write( "Using the quadratic formula again:\r<BR>\n" );
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document.write( "* a = (b ± √(b^2 + 4)) / 2\r<BR>\n" );
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document.write( "**4. Solve for x:**\r<BR>\n" );
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document.write( "* x = a + 4\r<BR>\n" );
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document.write( "**5. Numerical Solutions:**\r<BR>\n" );
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document.write( "Using numerical methods (like those used by the Python code), we find the real roots:\r<BR>\n" );
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document.write( "* x ≈ 2.445<BR>\n" );
document.write( "* x ≈ 4.690\r<BR>\n" );
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document.write( "**Final Answer:**\r<BR>\n" );
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document.write( "The real solutions are approximately x ≈ 2.445 and x ≈ 4.690.<BR>\n" );
document.write( " </SPAN>\n" );
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