document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111427>Question 111427</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two consecutive odd integers whose product is 79 more than their sum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81257 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=81257\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consecutive odd integers follow the form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B1 ALIGN=MIDDLE  ALT=\"2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3 ALIGN=MIDDLE  ALT=\"2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, etc\r<BR>\n" );
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document.write( "So the product of two such integers is 79 more than their sum translates to: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B1%29%282x%2B3%29=%282x%2B1%29%2B%282x%2B3%29%2B79 ALIGN=MIDDLE  ALT=\"%282x%2B1%29%282x%2B3%29=%282x%2B1%29%2B%282x%2B3%29%2B79\"   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=4x%5E2%2B8x%2B3=%282x%2B1%29%2B%282x%2B3%29%2B79 ALIGN=MIDDLE  ALT=\"4x%5E2%2B8x%2B3=%282x%2B1%29%2B%282x%2B3%29%2B79\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil the left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B8x%2B3=4x%2B83 ALIGN=MIDDLE  ALT=\"4x%5E2%2B8x%2B3=4x%2B83\"   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=4x%5E2%2B8x%2B3-4x-83=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B8x%2B3-4x-83=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get everything to one side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B4x-80=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B4x-80=0\"   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( "  <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;\" ><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=4%2Ax%5E2%2B4%2Ax-80=0 ALIGN=MIDDLE  ALT=\"4%2Ax%5E2%2B4%2Ax-80=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-80 ALIGN=MIDDLE  ALT=\"c=-80\"   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+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A4%2A-80+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A4%2A-80+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=4, b=4, and c=-80<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-4+%2B-+sqrt%28+16-4%2A4%2A-80+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+sqrt%28+16-4%2A4%2A-80+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 4 to get 16  <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-4+%2B-+sqrt%28+16%2B1280+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+sqrt%28+16%2B1280+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-80%2A4 ALIGN=MIDDLE  ALT=\"-4%2A-80%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1280 ALIGN=MIDDLE  ALT=\"1280\"   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+=+%28-4+%2B-+sqrt%28+1296+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+sqrt%28+1296+%29%29%2F%282%2A4%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+=+%28-4+%2B-+36%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+36%29%2F%282%2A4%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+=+%28-4+%2B-+36%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+36%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 4 to get 8<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+=+%28-4+%2B+36%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B+36%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-4+-+36%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%28-4+-+36%29%2F8\"   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=%28-4+%2B+36%29%2F8 ALIGN=MIDDLE  ALT=\"x=%28-4+%2B+36%29%2F8\"   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=32%2F8 ALIGN=MIDDLE  ALT=\"x=32%2F8\"   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=4 ALIGN=MIDDLE  ALT=\"x=4\"   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=4 ALIGN=MIDDLE  ALT=\"x=4\"   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=%28-4+-+36%29%2F8 ALIGN=MIDDLE  ALT=\"x=%28-4+-+36%29%2F8\"   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=-40%2F8 ALIGN=MIDDLE  ALT=\"x=-40%2F8\"   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=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   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( "  <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( "  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=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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 plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   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=2%284%29%2B1=9 ALIGN=MIDDLE  ALT=\"2%284%29%2B1=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%284%29%2B3=11 ALIGN=MIDDLE  ALT=\"2%284%29%2B3=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So one pair of numbers is 9,11\r<BR>\n" );
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document.write( "Now plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28-5%29%2B1=-9 ALIGN=MIDDLE  ALT=\"2%28-5%29%2B1=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28-5%29%2B3=-7 ALIGN=MIDDLE  ALT=\"2%28-5%29%2B3=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So another pair of numbers is -7,-9\r<BR>\n" );
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