document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=944234>Question 944234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two positive numbers that differ by eight and whose reciprocals sum is 1/6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #575697 by </B><A HREF=/tutors/aboutme.mpl?userid=macston><B>macston(5194)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=macston><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=macston><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=575697\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> X=one positive number, Y is the other positive number<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X-Y=8 ALIGN=MIDDLE  ALT=\"X-Y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >;  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2FX+%2B+1%2FY+=1%2F6 ALIGN=MIDDLE  ALT=\"1%2FX+%2B+1%2FY+=1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "X=Y+8 Add Y to each side of Equation 1 and substitute in Equation 2 <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28Y%2B8%29%2B1%2FY=1%2F6 ALIGN=MIDDLE  ALT=\"1%2F%28Y%2B8%29%2B1%2FY=1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply each side by 6<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%28Y%2B8%29%2B6%2FY=1 ALIGN=MIDDLE  ALT=\"6%2F%28Y%2B8%29%2B6%2FY=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 6/Y from each side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%28Y%2B8%29=1-6%2FY ALIGN=MIDDLE  ALT=\"6%2F%28Y%2B8%29=1-6%2FY\"   DISABLED_style= \"max-width:90%; height:auto;\" >  convert right side to fraction<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%28Y%2B8%29=1%28Y%2FY%29-6%2FY ALIGN=MIDDLE  ALT=\"6%2F%28Y%2B8%29=1%28Y%2FY%29-6%2FY\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%28Y%2B8%29=Y%2FY-6%2FY ALIGN=MIDDLE  ALT=\"6%2F%28Y%2B8%29=Y%2FY-6%2FY\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F%28Y%2B8%29=%28Y-6%29%2FY ALIGN=MIDDLE  ALT=\"6%2F%28Y%2B8%29=%28Y-6%29%2FY\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply each side by Y+8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=%28Y-6%29%28Y%2B8%29%2FY ALIGN=MIDDLE  ALT=\"6=%28Y-6%29%28Y%2B8%29%2FY\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply each side by Y<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6Y=%28Y-6%29%28Y%2B8%29 ALIGN=MIDDLE  ALT=\"6Y=%28Y-6%29%28Y%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply right side through<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6Y=Y%5E2%2B2Y-48 ALIGN=MIDDLE  ALT=\"6Y=Y%5E2%2B2Y-48\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 6Y from each side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=y%5E2-4Y-48 ALIGN=MIDDLE  ALT=\"0=y%5E2-4Y-48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=aY%5E2%2BbY%2Bc=0 ALIGN=MIDDLE  ALT=\"aY%5E2%2BbY%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1Y%5E2%2B-4Y%2B-48+=+0 ALIGN=MIDDLE  ALT=\"1Y%5E2%2B-4Y%2B-48+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"Y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-4%29%5E2-4%2A1%2A-48=208 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-4%29%5E2-4%2A1%2A-48=208\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=208 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--4%2B-sqrt%28+208+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--4%2B-sqrt%28+208+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+208+%29%29%2F2%5C1+=+9.21110255092798 ALIGN=MIDDLE  ALT=\"Y%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+208+%29%29%2F2%5C1+=+9.21110255092798\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%5B2%5D+=+%28-%28-4%29-sqrt%28+208+%29%29%2F2%5C1+=+-5.21110255092798 ALIGN=MIDDLE  ALT=\"Y%5B2%5D+=+%28-%28-4%29-sqrt%28+208+%29%29%2F2%5C1+=+-5.21110255092798\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1Y%5E2%2B-4Y%2B-48 ALIGN=MIDDLE  ALT=\"1Y%5E2%2B-4Y%2B-48\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1Y%5E2%2B-4Y%2B-48+=+1%28Y-9.21110255092798%29%2A%28Y--5.21110255092798%29 ALIGN=MIDDLE  ALT=\"1Y%5E2%2B-4Y%2B-48+=+1%28Y-9.21110255092798%29%2A%28Y--5.21110255092798%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 9.21110255092798, -5.21110255092798.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B-48+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B-48+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "The positive answer is 9.2111 so ANSWER 1 Y=9.2111<BR>\n" );
document.write( "X-Y=8<BR>\n" );
document.write( "X-9.2111=8 Add 9.2111 to each side<BR>\n" );
document.write( "X=17.2111 ANSWER 2 X=17.2111<BR>\n" );
document.write( "FINAL ANSWER THE TWO POSITIVE NUMBERS ARE 17.2111 and 9.2111<BR>\n" );
document.write( "CHECK <BR>\n" );
document.write( "1/X+1/Y=1/6<BR>\n" );
document.write( "1/17.2111+1/9.2111=1/6<BR>\n" );
document.write( "0.16667=1/6<BR>\n" );
document.write( "0.16667=0.16667\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );