document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209897>Question 1209897</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x and y be real numbers satisfying<BR>\n" );
document.write( "\frac{x^2y^2 - 1}{2y - 1} = 4x + y.<BR>\n" );
document.write( "Find the largest possible value of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851023 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=851023\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "(x^2y^2-1)/(2y-1) = 4x+y<BR>\n" );
document.write( "x^2y^2-1 = (4x+y)(2y-1)<BR>\n" );
document.write( "x^2y^2-1 = 8xy-4x+2y^2-y<BR>\n" );
document.write( "x^2y^2-1-8xy+4x-2y^2+y = 0<BR>\n" );
document.write( "(x^2-2)y^2+(-8x+1)y+4x-1 = 0\r<BR>\n" );
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document.write( "Compare that to ay^2+by+c = 0<BR>\n" );
document.write( "a = x^2-2<BR>\n" );
document.write( "b = -8x+1<BR>\n" );
document.write( "c = 4x-1<BR>\n" );
document.write( "The discriminant must be 0 or larger so that we end up with real number solutions for variable y.<BR>\n" );
document.write( "b^2-4ac >= 0<BR>\n" );
document.write( "(-8x+1)^2-4(x^2-2)(4x-1) >= 0<BR>\n" );
document.write( "-16x^3+68x^2+16x-7 >= 0\r<BR>\n" );
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document.write( "Use a graphing calculator such as <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> or <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to plot out the cubic curve f(x) = -16x^3+68x^2+16x-7 \r<BR>\n" );
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document.write( "The three approximate roots are <BR>\n" );
document.write( "p = -0.43067<BR>\n" );
document.write( "q = 0.22815<BR>\n" );
document.write( "r = 4.45252\r<BR>\n" );
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document.write( "Between roots q and r, we have -16x^3+68x^2+16x-7 above the x axis. <BR>\n" );
document.write( "At root r is when x is maxed out, such that the discriminant is zero and y is a real number.<BR>\n" );
document.write( "If x gets any larger, then the discriminant becomes negative and leads y to being a non-real complex number.\r<BR>\n" );
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document.write( "Answer: <font color=red>x = 4.45252 (approximate)</font><BR>\n" );
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