document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209657>Question 1209657</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the sum of all real numbers x satisfying\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2 - 6x + 4)(x^2 - 8x + 5) = 1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849746 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=849746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the sum of all real numbers x satisfying the given equation:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Rewrite the equation:**\r<BR>\n" );
document.write( "\n" );
document.write( "   (x² - 6x + 4)(x² - 8x + 5) = 1\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Substitution (optional):**\r<BR>\n" );
document.write( "\n" );
document.write( "   While not strictly necessary, you could substitute u = x² - 7x to try and simplify. However, it's easier to proceed without it.\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Expand and rearrange:**\r<BR>\n" );
document.write( "\n" );
document.write( "   Expanding the left side gives:\r<BR>\n" );
document.write( "\n" );
document.write( "   x⁴ - 8x³ + 5x² - 6x³ + 48x² - 30x + 4x² - 32x + 20 = 1<BR>\n" );
document.write( "   x⁴ - 14x³ + 57x² - 62x + 19 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Vieta's Formulas:**\r<BR>\n" );
document.write( "\n" );
document.write( "   For a quartic equation of the form ax⁴ + bx³ + cx² + dx + e = 0, the sum of the roots is given by -b/a. In our case, a = 1 and b = -14.\r<BR>\n" );
document.write( "\n" );
document.write( "   Therefore, the sum of all roots (real and complex) is -(-14)/1 = 14.\r<BR>\n" );
document.write( "\n" );
document.write( "5. **Consider the possibility of complex roots:**\r<BR>\n" );
document.write( "\n" );
document.write( "   A quartic equation can have up to four roots. Some of these roots could be complex. However, Vieta's formulas tell us the sum of *all* roots is 14.\r<BR>\n" );
document.write( "\n" );
document.write( "6. **Realizing that all roots must be real:**\r<BR>\n" );
document.write( "\n" );
document.write( "   Let f(x) = (x^2 - 6x + 4)(x^2 - 8x + 5) - 1.<BR>\n" );
document.write( "   Note that f(1) = (1-6+4)(1-8+5) - 1 = (-1)(-2) - 1 = 1 > 0.<BR>\n" );
document.write( "   f(2) = (4 - 12 + 4)(4 - 16 + 5) - 1 = (-4)(-7) - 1 = 27 > 0.<BR>\n" );
document.write( "   f(3) = (9 - 18 + 4)(9 - 24 + 5) - 1 = (-5)(-10) - 1 = 49 > 0.<BR>\n" );
document.write( "   f(4) = (16 - 24 + 4)(16 - 32 + 5) - 1 = (-4)(-11) - 1 = 43 > 0.<BR>\n" );
document.write( "   f(5) = (25 - 30 + 4)(25 - 40 + 5) - 1 = (-1)(-10) - 1 = 9 > 0.<BR>\n" );
document.write( "   f(6) = (36 - 36 + 4)(36 - 48 + 5) - 1 = 4(-7) - 1 = -29 < 0.<BR>\n" );
document.write( "   f(7) = (49 - 42 + 4)(49 - 56 + 5) - 1 = 11(-2) - 1 = -23 < 0.<BR>\n" );
document.write( "   f(8) = (64 - 48 + 4)(64 - 64 + 5) - 1 = 20(5) - 1 = 99 > 0.\r<BR>\n" );
document.write( "\n" );
document.write( "   Since f(x) is a continuous function, there must be a root between 6 and 8.<BR>\n" );
document.write( "   Also f(0) = 19 > 0.<BR>\n" );
document.write( "   Since the leading term is x^4, it goes to infinity as x goes to infinity and negative infinity.<BR>\n" );
document.write( "   Since f(1) > 0, f(6) < 0, f(8) > 0, there are two real roots between 1 and 8.<BR>\n" );
document.write( "   Since f(0) > 0 and f(x) goes to infinity as x goes to negative infinity, there must be a real root less than 0.<BR>\n" );
document.write( "   Since the sum of roots is 14, and there are 4 roots, all must be real.\r<BR>\n" );
document.write( "\n" );
document.write( "7. **Final Answer:**\r<BR>\n" );
document.write( "\n" );
document.write( "   The sum of all real roots is 14.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );