document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091579>Question 1091579</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The difference between two positive numbers is 18 and 4 times their GM is equal to 5 times their HM find the numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705993 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=705993\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) x - y = 18, with x, y > 0<BR>\n" );
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document.write( "GM is geometric mean, GM for x, y is defined as square root (x * y)<BR>\n" );
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document.write( "HM is harmonic mean, HM for x, y is defined as 2 / (1/x + 1/y) <BR>\n" );
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document.write( "2) 4 * square root(x * y) = 5 * (2 / (1/x + 1/y))<BR>\n" );
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document.write( "solve equation 1) for x<BR>\n" );
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document.write( "x = y + 18<BR>\n" );
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document.write( "substitute for x in equation 2<BR>\n" );
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document.write( "4 * square root((y+18) * y) = 5 * (2 / (1/(y+18) + 1/y))<BR>\n" );
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document.write( "4 * square root((y+18) * y) = 10 / (1/(y+18) + 1/y)<BR>\n" );
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document.write( "cross multiply the fractions<BR>\n" );
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document.write( "4 * square root((y+18) * y) * (1/(y+18) + 1/y) = 10<BR>\n" );
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document.write( "divide both sides of = by 4<BR>\n" );
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document.write( "square root((y+18) * y) * (1/(y+18) + 1/y) = 5/2<BR>\n" );
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document.write( "divide both sides of = by (1/(y+18) + 1/y)<BR>\n" );
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document.write( "square root((y+18) * y) = (5y * (y+18)) / (4 * (y+9))<BR>\n" );
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document.write( "square both sides of the =<BR>\n" );
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document.write( "(y+18) * y = (25y^2 * (y+18)^2) / (16 * (y+9)^2<BR>\n" );
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document.write( "cross multiply the fractions<BR>\n" );
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document.write( "16y * (y+18) * (y+9)^2 = 25y^2 * (y+18)^2<BR>\n" );
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document.write( "write the left side of the = in standard form<BR>\n" );
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document.write( "16y^4 +576y^3 +6480y^2 +23328y = 25y^2 * (y+18)^2<BR>\n" );
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document.write( "write the right side of the = in standard form<BR>\n" );
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document.write( "16y^4 +576y^3 +6480y^2 +23328y = 25y^4 +900y^3 +8100y^2<BR>\n" );
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document.write( "consolidate the terms to the left side of =<BR>\n" );
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document.write( "-9y^4 -324y^3 -1620y^2 +23328y = 0<BR>\n" );
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document.write( "this factors into<BR>\n" );
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document.write( "-9y * (y-6) * (y+18) * (y+24) = 0<BR>\n" );
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document.write( "divide both sides of = by -9<BR>\n" );
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document.write( "our solutions for y are 0, 6, -18, -24<BR>\n" );
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document.write( "we reject solutions 0, -18, -24 since y > 0 <BR>\n" );
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document.write( "x = 6 + 18 = 24<BR>\n" );
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document.write( "x = 24 and y = 6<BR>\n" );
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document.write( " </SPAN>\n" );
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