document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211832>Question 211832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The product of two consecutive odd integers is 1 less than four times their sum.<BR>\n" );
document.write( "Find the two integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #160648 by </B><A HREF=/tutors/aboutme.mpl?userid=drj><B>drj(1380)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drj><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=drj><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=160648\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The product of two consecutive odd integers is 1 less than four times their sum.<BR>\n" );
document.write( "Find the two integers.<BR>\n" );
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document.write( "Step 1.  Let n be one odd integer.\r<BR>\n" );
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document.write( "Step 2.  Let n+2 be the next consecutive odd integer.\r<BR>\n" );
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document.write( "Step 3.  Translate problem statement into an equation such than n(n+2)=4(n+n+2)-1 since product is 1 less than four times their sum\r<BR>\n" );
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document.write( "where n(n+2) is the product of the two consecutive integers and (n+n+2) is the sum of the two integers\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B2%29=4%28n%2Bn%2B2%29-1 ALIGN=MIDDLE  ALT=\"n%28n%2B2%29=4%28n%2Bn%2B2%29-1\"   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=n%5E2%2B2n=4%282n%2B2%29-1 ALIGN=MIDDLE  ALT=\"n%5E2%2B2n=4%282n%2B2%29-1\"   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=n%5E2%2B2n=8n%2B8-1 ALIGN=MIDDLE  ALT=\"n%5E2%2B2n=8n%2B8-1\"   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=n%5E2%2B2n=8n%2B7 ALIGN=MIDDLE  ALT=\"n%5E2%2B2n=8n%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 4.  Simplify to obtain a quadratic equation.  Subtract 8n+7 from both sides of equation to get the right side of equation to be zero.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2B2n-8n-7=8n%2B7-8n-7 ALIGN=MIDDLE  ALT=\"n%5E2%2B2n-8n-7=8n%2B7-8n-7\"   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=n%5E2-6n-7=0 ALIGN=MIDDLE  ALT=\"n%5E2-6n-7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 5.  Now we have a quadratic equation.  So use quadratic formula given as \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where\r<BR>\n" );
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document.write( "a=1, b=-6 and  c=-7\r<BR>\n" );
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document.write( "The following steps solves the quadratic equation\r<BR>\n" );
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document.write( "<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=an%5E2%2Bbn%2Bc=0 ALIGN=MIDDLE  ALT=\"an%5E2%2Bbn%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1n%5E2%2B-6n%2B-7+=+0 ALIGN=MIDDLE  ALT=\"1n%5E2%2B-6n%2B-7+=+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=n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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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=%28-6%29%5E2-4%2A1%2A-7=64 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-6%29%5E2-4%2A1%2A-7=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  Discriminant d=64 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--6%2B-sqrt%28+64+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--6%2B-sqrt%28+64+%29%29%2F2%5Ca\"   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=n%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+7 ALIGN=MIDDLE  ALT=\"n%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5B2%5D+=+%28-%28-6%29-sqrt%28+64+%29%29%2F2%5C1+=+-1 ALIGN=MIDDLE  ALT=\"n%5B2%5D+=+%28-%28-6%29-sqrt%28+64+%29%29%2F2%5C1+=+-1\"   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=1n%5E2%2B-6n%2B-7 ALIGN=MIDDLE  ALT=\"1n%5E2%2B-6n%2B-7\"   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=1n%5E2%2B-6n%2B-7+=+1%28n-7%29%2A%28n--1%29 ALIGN=MIDDLE  ALT=\"1n%5E2%2B-6n%2B-7+=+1%28n-7%29%2A%28n--1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 7, -1.\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-6%2Ax%2B-7+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B-7+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Step 6.  ANSWER:  Based on the above steps, choose 7 since it is positive.  The integers are then 7 and 9.\r<BR>\n" );
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document.write( "Step 7.  Verify the answer using n(n+2)=4(n+n+2)+1 in Step 3.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=63=4%2816%29-1 ALIGN=MIDDLE  ALT=\"63=4%2816%29-1\"   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=63=63 ALIGN=MIDDLE  ALT=\"63=63\"   DISABLED_style= \"max-width:90%; height:auto;\" >  So it checks out.\r<BR>\n" );
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document.write( "I hope the above steps were helpful. \r<BR>\n" );
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document.write( "And good luck in your studies!\r<BR>\n" );
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document.write( "Respectfully,<BR>\n" );
document.write( "Dr J\r<BR>\n" );
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document.write( "For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra.\r<BR>\n" );
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