document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541670>Question 541670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of a number and its reciprocal is 85/42. Find the number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354259 by </B><A HREF=/tutors/aboutme.mpl?userid=neatmath><B>neatmath(302)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=neatmath><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=neatmath><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=354259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><font face=\"Tahoma\">Let's first set up an equation using x as our variable:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Fx=85%2F42 ALIGN=MIDDLE  ALT=\"x%2B1%2Fx=85%2F42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Then we just need to solve for x:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42x%2B42%2Fx=85 ALIGN=MIDDLE  ALT=\"42x%2B42%2Fx=85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42x%5E2%2B42x%2Fx=85x ALIGN=MIDDLE  ALT=\"42x%5E2%2B42x%2Fx=85x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42x%5E2%2B42=85x ALIGN=MIDDLE  ALT=\"42x%5E2%2B42=85x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42x%5E2-85x%2B42=0 ALIGN=MIDDLE  ALT=\"42x%5E2-85x%2B42=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Then we can use the quadratic formula to solve for x:<br>\r<BR>\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=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42x%5E2%2B-85x%2B42+=+0 ALIGN=MIDDLE  ALT=\"42x%5E2%2B-85x%2B42+=+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=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%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=%28-85%29%5E2-4%2A42%2A42=169 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-85%29%5E2-4%2A42%2A42=169\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=169 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--85%2B-sqrt%28+169+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--85%2B-sqrt%28+169+%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=x%5B1%5D+=+%28-%28-85%29%2Bsqrt%28+169+%29%29%2F2%5C42+=+1.16666666666667 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-85%29%2Bsqrt%28+169+%29%29%2F2%5C42+=+1.16666666666667\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-85%29-sqrt%28+169+%29%29%2F2%5C42+=+0.857142857142857 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-85%29-sqrt%28+169+%29%29%2F2%5C42+=+0.857142857142857\"   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=42x%5E2%2B-85x%2B42 ALIGN=MIDDLE  ALT=\"42x%5E2%2B-85x%2B42\"   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=42x%5E2%2B-85x%2B42+=+42%28x-1.16666666666667%29%2A%28x-0.857142857142857%29 ALIGN=MIDDLE  ALT=\"42x%5E2%2B-85x%2B42+=+42%28x-1.16666666666667%29%2A%28x-0.857142857142857%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.16666666666667, 0.857142857142857.\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+42%2Ax%5E2%2B-85%2Ax%2B42+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+42%2Ax%5E2%2B-85%2Ax%2B42+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "<br>\r<BR>\n" );
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document.write( "This gives us a good approximation to work with.<br>\r<BR>\n" );
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document.write( "Using an online factoring calculator, we find that our equation factors as:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287x-6%29%286x-7%29=0 ALIGN=MIDDLE  ALT=\"%287x-6%29%286x-7%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "From this we can easily see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6%2F7 ALIGN=MIDDLE  ALT=\"x=6%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7%2F6 ALIGN=MIDDLE  ALT=\"x=7%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "It would take us a BIT of time to factor that equation properly and get the exact answer.<br>\r<BR>\n" );
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document.write( "I'm trying to think of other ways off the top of my head to solve this.<br>\r<BR>\n" );
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document.write( "I hope this helps!<br> </SPAN>\n" );
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