document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=524>Question 524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I factor 6x^2-7x-4? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #177 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1806)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=177\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face = \"courier new\">How do I factor 6x^2-7x-4<BR>\n" );
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document.write( "Make a list of all the possible ways for <BR>\n" );
document.write( "the FIRSTS to have product 6x²<BR>\n" );
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document.write( "(1x ` )(6x ` )<BR>\n" );
document.write( "(2x ` )(3x ` )<BR>\n" );
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document.write( "Make a list of all the ways for the<BR>\n" );
document.write( "LASTS to have product 4<BR>\n" );
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document.write( "( ` 1)( ` 4)<BR>\n" );
document.write( "( ` 2)( ` 2) <BR>\n" );
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document.write( "Pick one of the above lists to extend to <BR>\n" );
document.write( "include all the \"reversals\", i.e., the <BR>\n" );
document.write( "\"reversal of \"1×4\" is \"4×1\"<BR>\n" );
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document.write( "I pick the list of factors of 4, because <BR>\n" );
document.write( "all I have to do is add on the reversal <BR>\n" );
document.write( "\"4×1\".  So we have these lists of <BR>\n" );
document.write( "possibilities for the FIRSTS:<BR>\n" );
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document.write( "(1x ` )(6x ` )<BR>\n" );
document.write( "(2x ` )(3x ` )<BR>\n" );
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document.write( "and these possibilities for lasts:<BR>\n" );
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document.write( "( ` ` 1)( ` ` 4)<BR>\n" );
document.write( "( ` ` 2)( ` ` 2)<BR>\n" );
document.write( "( ` ` 4)( ` ` 1)<BR>\n" );
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document.write( "Now put each of the first two with each of<BR>\n" );
document.write( "the last three<BR>\n" );
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document.write( "(1x ` 1)(6x ` 4)<BR>\n" );
document.write( "(1x ` 2)(6x ` 2)<BR>\n" );
document.write( "(1x ` 4)(6x ` 1)<BR>\n" );
document.write( "(2x ` 1)(3x ` 4)<BR>\n" );
document.write( "(2x ` 2)(3x ` 2)<BR>\n" );
document.write( "(2x ` 4)(3x ` 1)<BR>\n" );
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document.write( "Strike through any of these which contains a<BR>\n" );
document.write( "parenthetical binomial from which a common <BR>\n" );
document.write( "factor can be taken out <BR>\n" );
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document.write( "We strike through (eliminate) the first one <BR>\n" );
document.write( "because 2 can be taken out of (6x ` 4)<BR>\n" );
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document.write( "We strike through (eliminate) the second one<BR>\n" );
document.write( "because 2 can be taken out of (6x ` 2)<BR>\n" );
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document.write( "We strike through (eliminate) the fifth one <BR>\n" );
document.write( "because 2 can be taken out of (2x ` 2)<BR>\n" );
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document.write( "We strike through (eliminate) the sixth one <BR>\n" );
document.write( "because 2 can be taken out of (2x ` 4) <BR>\n" );
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document.write( "So now our list is<BR>\n" );
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document.write( "<s>(1x ` 1)(6x ` 4)</s><BR>\n" );
document.write( "<s>(1x ` 2)(6x ` 2)</s><BR>\n" );
document.write( "(1x ` 4)(6x ` 1)<BR>\n" );
document.write( "(2x ` 1)(3x ` 4)<BR>\n" );
document.write( "<s>(2x ` 2)(3x ` 2)</s><BR>\n" );
document.write( "<s>(2x ` 4)(3x ` 1)</s><BR>\n" );
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document.write( "So all we have left are these two.<BR>\n" );
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document.write( "(1x ` 4)(6x ` 1)<BR>\n" );
document.write( "(2x ` 1)(3x ` 4)<BR>\n" );
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document.write( "Since the last term of the trinomial <BR>\n" );
document.write( "6x^2-7x-4 is negative, the signs are different, <BR>\n" );
document.write( "so we need only try<BR>\n" );
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document.write( "(1x + 4)(6x - 1)<BR>\n" );
document.write( "(2x + 1)(3x - 4)<BR>\n" );
document.write( "(1x - 4)(6x + 1)<BR>\n" );
document.write( "(2x - 1)(3x + 4)<BR>\n" );
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document.write( "Multiplying them out<BR>\n" );
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document.write( "(1x + 4)(6x - 1) = 6x² + 23x - 4<BR>\n" );
document.write( "(2x + 1)(3x - 4) = 6x² - 5x - 4<BR>\n" );
document.write( "(1x - 4)(6x + 1) = 6x² - 23x - 4<BR>\n" );
document.write( "(2x - 1)(3x + 4) = 6x² + 5x - 4<BR>\n" );
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document.write( "None of those are 6x²-7x-4, so 6x²-7x-4 is <BR>\n" );
document.write( "prime and thus will not factor.<BR>\n" );
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document.write( "Edwin </SPAN>\n" );
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