document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92349>Question 92349</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all positive 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 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67307 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=67307\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x^2 + x - k----Quadratic in standard form where:\r<BR>\n" );
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document.write( "A=1<BR>\n" );
document.write( "B=1<BR>\n" );
document.write( "C=k\r<BR>\n" );
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document.write( "When we have a quadratic where the A coefficient is equal to 1, then the factors of C must add up to the B coefficient.  For example, lets assume that the factors for your equation are:\r<BR>\n" );
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document.write( "(x+a)(x-b)=0 \r<BR>\n" );
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document.write( "Expand using the FOIL concept(First, Outer, Inner, Last)\r<BR>\n" );
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document.write( "x^2-bx+ax-ab=0  rearrange\r<BR>\n" );
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document.write( "x^2+(a-b)x-ab=0--------------eq1\r<BR>\n" );
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document.write( "Now in your problem, clearly (a-b) must equal 1\r<BR>\n" );
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document.write( "So, if a-b=1 in eq1, then b=(a-1) \r<BR>\n" );
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document.write( "Substitute b=(a-1) into eq1 and rewrite it\r<BR>\n" );
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document.write( "x^2+(a-(a-1))x-a(a-1)=0  simplify\r<BR>\n" );
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document.write( "x^2+(a-a+1)x-a(a-1)=0 or, further simplifying\r<BR>\n" );
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document.write( "x^2+x-a(a-1)=0-----------eq2  Now we know that k=a(a-1) where a has to be greater than or equal to zero.  \r<BR>\n" );
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document.write( "Hope this helps--------------ptaylor\r<BR>\n" );
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