document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=27864>Question 27864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all positie values for K for which each of the following can be factored.\r<BR>\n" );
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document.write( "x^2 + x - k\r<BR>\n" );
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document.write( "Thank you for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15179 by </B><A HREF=/tutors/aboutme.mpl?userid=bmauger><B>bmauger(101)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bmauger><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=15179\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It can only be factored if the discriminant is positive and a perfect square. The discriminant is defined by:<BR>\n" );
document.write( "<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;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "For your problem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+k ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+k\"   DISABLED_style= \"max-width:90%; height:auto;\" > a=1, b=1 and c=-k<BR>\n" );
document.write( "so the descriminant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B4k ALIGN=MIDDLE  ALT=\"1%2B4k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So any number that when multiplied by 4 is one less than a square number would work. Such as:<BR>\n" );
document.write( "0 --> 1+4*0=1 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx=%28x%2B1%29%28x%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx=%28x%2B1%29%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2 --> 1+4*2=9 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-2=%28x%2B2%29%28x-1%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-2=%28x%2B2%29%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "6 --> 1+4*6=25 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-6=%28x%2B3%29%28x-2%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-6=%28x%2B3%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "12 --> 1+4*12=49 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-12=%28x%2B4%29%28x-3%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-12=%28x%2B4%29%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "20 --> 1+4*20=81 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-20=%28x%2B5%29%28x-4%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-20=%28x%2B5%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "30 --> 1+4*30=121 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-30=%28x%2B6%29%28x-5%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-30=%28x%2B6%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "42 --> 1+4*42=169 --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-42=%28x%2B7%29%28x-6%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-42=%28x%2B7%29%28x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Starting to see some paterns, no doubt?...<BR>\n" );
document.write( "So there's an infinite number of k's that'll work. To find the next value in the pattern, square each odd number, subtract one, and divide by 4.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%5E2-1%29%2F4=0 ALIGN=MIDDLE  ALT=\"%281%5E2-1%29%2F4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5E2-1%29%2F4=2 ALIGN=MIDDLE  ALT=\"%283%5E2-1%29%2F4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%5E2-1%29%2F4=6 ALIGN=MIDDLE  ALT=\"%285%5E2-1%29%2F4=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E2-1%29%2F4=12 ALIGN=MIDDLE  ALT=\"%287%5E2-1%29%2F4=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%5E2-1%29%2F4=20 ALIGN=MIDDLE  ALT=\"%289%5E2-1%29%2F4=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811%5E2-1%29%2F4=30 ALIGN=MIDDLE  ALT=\"%2811%5E2-1%29%2F4=30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2813%5E2-1%29%2F4=42 ALIGN=MIDDLE  ALT=\"%2813%5E2-1%29%2F4=42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%5E2-1%29%2F4=56 ALIGN=MIDDLE  ALT=\"%2815%5E2-1%29%2F4=56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2817%5E2-1%29%2F4=72 ALIGN=MIDDLE  ALT=\"%2817%5E2-1%29%2F4=72\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer your question? </SPAN>\n" );
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