document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=294753>Question 294753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two consecutive odd integers such that the sum of their reciprocals is 12/35.I have been trying for three days to write a problem out for this question and I am for some reason not able to come up with an answer. Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #212542 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=212542\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consecutive odd integers follow the pattern: x, x+2, ...\r<BR>\n" );
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document.write( "So adding their reciprocals to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F35 ALIGN=MIDDLE  ALT=\"12%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" > means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%2B1%2F%28x%2B2%29=12%2F35 ALIGN=MIDDLE  ALT=\"1%2Fx%2B1%2F%28x%2B2%29=12%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%2B1%2F%28x%2B2%29=12%2F35 ALIGN=MIDDLE  ALT=\"1%2Fx%2B1%2F%28x%2B2%29=12%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%28x%2B2%29%2B35x=12x%28x%2B2%29 ALIGN=MIDDLE  ALT=\"35%28x%2B2%29%2B35x=12x%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35x%28x%2B2%29 ALIGN=MIDDLE  ALT=\"35x%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35x%2B70%2B35x=12x%5E2%2B24x ALIGN=MIDDLE  ALT=\"35x%2B70%2B35x=12x%5E2%2B24x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70x%2B70=12x%5E2%2B24x ALIGN=MIDDLE  ALT=\"70x%2B70=12x%5E2%2B24x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=12x%5E2%2B24x-70x-70 ALIGN=MIDDLE  ALT=\"0=12x%5E2%2B24x-70x-70\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get every term to one side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=12x%5E2-46x-70 ALIGN=MIDDLE  ALT=\"0=12x%5E2-46x-70\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\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_formula.solver>Quadratic Formula</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Let's use the quadratic formula to solve for x:<BR><BR>\n" );
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document.write( "  Starting with the general quadratic<BR>\n" );
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document.write( "  the general solution using the quadratic equation is:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2Ax%5E2-46%2Ax-70=0 ALIGN=MIDDLE  ALT=\"12%2Ax%5E2-46%2Ax-70=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=12 ALIGN=MIDDLE  ALT=\"a=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-46 ALIGN=MIDDLE  ALT=\"b=-46\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-70 ALIGN=MIDDLE  ALT=\"c=-70\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28--46+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%28--46+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=12, b=-46, and c=-70<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+sqrt%28+%28-46%29%5E2-4%2A12%2A-70+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -46 to get 46<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+sqrt%28+2116-4%2A12%2A-70+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+sqrt%28+2116-4%2A12%2A-70+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -46 to get 2116  (note: remember when you square -46, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-46%29%5E2=-46%2A-46=2116 ALIGN=MIDDLE  ALT=\"%28-46%29%5E2=-46%2A-46=2116\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+sqrt%28+2116%2B3360+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+sqrt%28+2116%2B3360+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-70%2A12 ALIGN=MIDDLE  ALT=\"-4%2A-70%2A12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3360 ALIGN=MIDDLE  ALT=\"3360\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+sqrt%28+5476+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+sqrt%28+5476+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+74%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+74%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B-+74%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B-+74%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 12 to get 24<BR>\n" );
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document.write( "  So now the expression breaks down into two parts<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+%2B+74%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%2846+%2B+74%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2846+-+74%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%2846+-+74%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Lets look at the first part:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2846+%2B+74%29%2F24 ALIGN=MIDDLE  ALT=\"x=%2846+%2B+74%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=120%2F24 ALIGN=MIDDLE  ALT=\"x=120%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide<BR>\n" );
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document.write( "  So one answer is<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now lets look at the second part:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2846+-+74%29%2F24 ALIGN=MIDDLE  ALT=\"x=%2846+-+74%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-28%2F24 ALIGN=MIDDLE  ALT=\"x=-28%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator<BR>\n" );
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document.write( "  <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;\" > Divide<BR>\n" );
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document.write( "  So another answer is<BR>\n" );
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document.write( "  So our solutions are:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   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>\n" );
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document.write( "Since we're only looking for integer values of 'x', we can ignore <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;\" >.\r<BR>\n" );
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document.write( "So the only answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that the two consecutive odd integers are 5 and 7.\r<BR>\n" );
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document.write( "Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F5%2B1%2F7=7%2F35%2B5%2F35=%287%2B5%29%2F35=12%2F35 ALIGN=MIDDLE  ALT=\"1%2F5%2B1%2F7=7%2F35%2B5%2F35=%287%2B5%29%2F35=12%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" > which confirms our answer. </SPAN>\n" );
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