document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027738>Question 1027738</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve using quadratic formula.\r<BR>\n" );
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document.write( "1 over X^2 minus 3 equals 8 over x\r<BR>\n" );
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document.write( "The solution is -4 over 3 (plus or minus) the square root of 19 over 3. I don't understand how. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642940 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=642940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> me neither.\r<BR>\n" );
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document.write( "i graphed the factors and got the following graph.\r<BR>\n" );
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document.write( "the factors are (3x + 4 - sqrt(19)) * (3x + 4 + sqrt(19))\r<BR>\n" );
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document.write( "when you multiply those factors out, you get 9x^2 + 24x - 3.\r<BR>\n" );
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document.write( "i took that and then did the following:\r<BR>\n" );
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document.write( "start with 9x^2 + 24x - 3\r<BR>\n" );
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document.write( "subtract 24x from both sides to get:\r<BR>\n" );
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document.write( "9x^2 - 3 = -24x\r<BR>\n" );
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document.write( "factor the left side of the equation to get:\r<BR>\n" );
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document.write( "3 * (3x^2 - 1) = -24x\r<BR>\n" );
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document.write( "divide both sides of the equation to get:\r<BR>\n" );
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document.write( "3x^2 - 1 = -8x\r<BR>\n" );
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document.write( "divide both sides of the equation by 8x to get:\r<BR>\n" );
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document.write( "(3x^2 - 1) / 8x = -1\r<BR>\n" );
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document.write( "divide both sides of the equation by (3x^2 - 1) to get:\r<BR>\n" );
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document.write( "1 / 8x = -1 / (3x^2 - 1)\r<BR>\n" );
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document.write( "i then added -1 / (3x^2 - 1) to both sides of this equation to get:\r<BR>\n" );
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document.write( "1 / 8x + 1 / (3x^2 - 1) = 0\r<BR>\n" );
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document.write( "the roots of this equation are the same roots as the equation of 9x^2 + 24x - 3 = 0.\r<BR>\n" );
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document.write( "i graphed 3 equations to show you that the roots are the same for all 3.\r<BR>\n" );
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document.write( "the first 2 equations i graphed are:\r<BR>\n" );
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document.write( "y = (3x + 4 - sqrt(19)) * (3x + 4 + sqrt(19))\r<BR>\n" );
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document.write( "y = 9x^2 + 24x - 3.\r<BR>\n" );
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document.write( "these 2 equations are equivalent and show up as the same line on the graph.\r<BR>\n" );
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document.write( "the roots are x = -2.786 and x = .12\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040206.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the other equation i graphed is:\r<BR>\n" );
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document.write( "y = 1 / 8x + 1 / (3x^2 - 1)\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040205.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "while this graph looks very different, the roots are the same.\r<BR>\n" );
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document.write( "the closest i could get to what you were showing with the roots you showed is:\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "9x^2 + 24x - 3 = 0\r<BR>\n" );
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document.write( "subtract 24x from both sides of the equation to get:\r<BR>\n" );
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document.write( "9x^2 - 3 = -24x\r<BR>\n" );
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document.write( "factor out the 3 on the left hand side to get:\r<BR>\n" );
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document.write( "3 * (3x^2 - 1) = -24x\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "3x^2 - 1 = -8x\r<BR>\n" );
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document.write( "divide both sides of the equation by x and divide both sides of the equation by (3x^2 - 1) to get:\r<BR>\n" );
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document.write( "1/x = -8 / (3x^2 - 1)\r<BR>\n" );
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document.write( "that's the closest i could get to what you were showing.\r<BR>\n" );
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document.write( "it's not it.\r<BR>\n" );
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document.write( "what you are showing does not result in the roots you are showing.\r<BR>\n" );
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document.write( "i took the equation of 1/x = -8 / (3x^2 - 1) and added the right side of the equation to to both sides of the equation to get:\r<BR>\n" );
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document.write( "1/x + 8/(3x^2-1) = 0\r<BR>\n" );
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document.write( "the roots of that equation are the same, even though that equation looks different also.\r<BR>\n" );
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document.write( "the graph of y = 1/x + 8/(3x^2-1) is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040207.jpg\" alt=$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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