document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948331>Question 948331</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the product of two consecutive integers is 25 less than 5 times 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 #578826 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=578826\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> X=first integer; X+1=second integer<BR>\n" );
document.write( "(X)(X+1)=5(X+(X+1))-25<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2%2BX=5X%2B5X%2B5-25 ALIGN=MIDDLE  ALT=\"X%5E2%2BX=5X%2B5X%2B5-25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2%2BX=10X-20 ALIGN=MIDDLE  ALT=\"X%5E2%2BX=10X-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 10X from each side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2-9X=-20 ALIGN=MIDDLE  ALT=\"X%5E2-9X=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 20 to each side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2-9X%2B20=0 ALIGN=MIDDLE  ALT=\"X%5E2-9X%2B20=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=1X%5E2%2B-9X%2B20+=+0 ALIGN=MIDDLE  ALT=\"1X%5E2%2B-9X%2B20+=+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=%28-9%29%5E2-4%2A1%2A20=1 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-9%29%5E2-4%2A1%2A20=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1 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--9%2B-sqrt%28+1+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--9%2B-sqrt%28+1+%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-%28-9%29%2Bsqrt%28+1+%29%29%2F2%5C1+=+5 ALIGN=MIDDLE  ALT=\"X%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+1+%29%29%2F2%5C1+=+5\"   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-%28-9%29-sqrt%28+1+%29%29%2F2%5C1+=+4 ALIGN=MIDDLE  ALT=\"X%5B2%5D+=+%28-%28-9%29-sqrt%28+1+%29%29%2F2%5C1+=+4\"   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=1X%5E2%2B-9X%2B20 ALIGN=MIDDLE  ALT=\"1X%5E2%2B-9X%2B20\"   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=1X%5E2%2B-9X%2B20+=+1%28X-5%29%2A%28X-4%29 ALIGN=MIDDLE  ALT=\"1X%5E2%2B-9X%2B20+=+1%28X-5%29%2A%28X-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 5, 4.\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-9%2Ax%2B20+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-9%2Ax%2B20+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "ANSWERS form X are 4 and 5<BR>\n" );
document.write( "CHECK<BR>\n" );
document.write( "For X=4, X+1=5<BR>\n" );
document.write( "4(5)=5(4+5)-25<BR>\n" );
document.write( "20=45-25<BR>\n" );
document.write( "20=20 True, so 4 is a correct answer. ANSWER 1: One pair of consecutive integers is 4,5<BR>\n" );
document.write( "For X=5, X+1=6<BR>\n" );
document.write( "5(6)=5(5+6)-25<BR>\n" );
document.write( "30=55-25<BR>\n" );
document.write( "30=30 So 5 is also a correct answer. ANSWER 2: Another pair of consecutive integers is 5,6.<BR>\n" );
document.write( "So there are two pairs of consecutive integers that fulfill these requirements, (4,5) and (5,6). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );