document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=352757>Question 352757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are three consecutive numbers that have a sum which is 1/5 of their product? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #252148 by </B><A HREF=/tutors/aboutme.mpl?userid=jrfrunner><B>jrfrunner(365)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jrfrunner><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=jrfrunner><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=252148\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 3 consecutive numbers:  x, x+1, x+2<BR>\n" );
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document.write( "sum is 1/5 their product:  x+(x+1)+(x+2)=1/5[x*(x+1)*(x+2)]<BR>\n" );
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document.write( " x+(x+1)+(x+2)=1/5[x*(x+1)*(x+2)]<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3=%28x%5E2%2Bx%29%28x%2B2%29%2F5 ALIGN=MIDDLE  ALT=\"3x%2B3=%28x%5E2%2Bx%29%28x%2B2%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%2B15=%28x%5E2%2Bx%29%2A%28x%2B2%29=x%5E3%2B2x%5E2%2Bx%5E2%2B2x ALIGN=MIDDLE  ALT=\"15x%2B15=%28x%5E2%2Bx%29%2A%28x%2B2%29=x%5E3%2B2x%5E2%2Bx%5E2%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=x%5E3%2B3x%5E2-13x-15 ALIGN=MIDDLE  ALT=\"0=x%5E3%2B3x%5E2-13x-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since highest power is cubed, there are a total of 3 roots<BR>\n" );
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document.write( "by descartes sign test,  there is one positive real root<BR>\n" );
document.write( "possible real roots -+1,-+3,-+5,-+15<BR>\n" );
document.write( "actual roots will rarely be in the extremes, so try near the middle, try 3<BR>\n" );
document.write( "via synthetic division.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2-13x-15 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2-13x-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > divided by root 3 yields <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%2B5 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "in other words<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2-13x-15=%28x-3%29%2A%28x%5E2%2B6x%2B5%29 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2-13x-15=%28x-3%29%2A%28x%5E2%2B6x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%2B5 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(x+5)*(x+1)<BR>\n" );
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document.write( "Therefore: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2-13x-15=%28x-3%29%2A%28x%2B5%29%2A%28x%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2-13x-15=%28x-3%29%2A%28x%2B5%29%2A%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "with potentially 3 solutions<BR>\n" );
document.write( "if x=3, then x+1=4 and x+2=5<BR>\n" );
document.write( "if x=-5 then x+1=-4 and x+2=-3<BR>\n" );
document.write( "if x=-1 then x+1=0 and x+2=1\r<BR>\n" );
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document.write( "(3,4,5) or (-5,-4,-3) or (-1,0,1)<BR>\n" );
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document.write( "validate each set of solutions<BR>\n" );
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document.write( "(3,4,5)<BR>\n" );
document.write( "sum=3+4+5=12<BR>\n" );
document.write( "1/5 product= 1/5*(3*4*5)=60/5=12<BR>\n" );
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document.write( "(-5,-4,-3)<BR>\n" );
document.write( "sum=-5-4-3=-12<BR>\n" );
document.write( "1/5 product= 1/5*(-5)*(-4)*(-3)=-60/5=-12<BR>\n" );
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document.write( "(1,0,-1)<BR>\n" );
document.write( "sum=1+0-1=0<BR>\n" );
document.write( "1/5 product = 1/5*(1)*(0)*(-1)=0/5=0<BR>\n" );
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document.write( "so there exists more than one solution. All three of these solutions work. </SPAN>\n" );
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