document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209109>Question 1209109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all values of x such that<BR>\n" );
document.write( "9(x + 3) = -\frac{8}{x} + \frac{2}{x - 5}.<BR>\n" );
document.write( "If you find more than one value, then list your solutions, separated by commas. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847826 by </B><A HREF=/tutors/aboutme.mpl?userid=yurtman><B>yurtman(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=yurtman><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=847826\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this equation, we'll first clear the denominators by multiplying both sides by $x(x-5)$:\r<BR>\n" );
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document.write( "$$9(x+3)x(x-5) = -8(x-5) + 2x$$\r<BR>\n" );
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document.write( "Expand the left side:\r<BR>\n" );
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document.write( "$$9x^3 - 45x^2 + 27x^2 - 135x = -8x + 40 + 2x$$\r<BR>\n" );
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document.write( "Combine like terms:\r<BR>\n" );
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document.write( "$$9x^3 - 18x^2 - 127x - 40 = 0$$\r<BR>\n" );
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document.write( "Unfortunately, this is a cubic equation, and there's no simple algebraic method to solve it directly. We can use numerical methods or software tools to find approximate solutions. \r<BR>\n" );
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document.write( "However, we can try to factor the equation to see if we can find any rational solutions. By using the Rational Root Theorem, we can test potential rational roots.\r<BR>\n" );
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document.write( "By trial and error, we find that $x = -1$ is a solution. This means that $(x+1)$ is a factor of the cubic polynomial. We can perform polynomial long division to find the other factor:\r<BR>\n" );
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document.write( "$$9x^3 - 18x^2 - 127x - 40 = (x+1)(9x^2 - 27x - 40)$$\r<BR>\n" );
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document.write( "Now, we can solve the quadratic equation $9x^2 - 27x - 40 = 0$ using the quadratic formula:\r<BR>\n" );
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document.write( "$$x = \frac{-(-27) \pm \sqrt{(-27)^2 - 4(9)(-40)}}{2(9)}$$\r<BR>\n" );
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document.write( "Simplifying:\r<BR>\n" );
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document.write( "$$x = \frac{27 \pm \sqrt{2001}}{18}$$\r<BR>\n" );
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document.write( "So, the solutions to the original equation are:\r<BR>\n" );
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document.write( "$$x = -1, \frac{27 + \sqrt{2001}}{18}, \frac{27 - \sqrt{2001}}{18}$$<BR>\n" );
document.write( " </SPAN>\n" );
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