document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117766>Question 117766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you factor bē -20b+51 ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85741 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=85741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-20b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-20b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=51 ALIGN=MIDDLE  ALT=\"51\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 51 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 51 to get 51. Now what two numbers multiply to 51 and add to the  middle coefficient -20? Let's list all of the factors of 51:\r<BR>\n" );
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document.write( "Factors of 51:\r<BR>\n" );
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document.write( "1,3,17,51\r<BR>\n" );
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document.write( "-1,-3,-17,-51 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to 51\r<BR>\n" );
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document.write( "1*51\r<BR>\n" );
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document.write( "3*17\r<BR>\n" );
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document.write( "(-1)*(-51)\r<BR>\n" );
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document.write( "(-3)*(-17)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -20\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">51</TD><TD >1+51=52</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">17</TD><TD >3+17=20</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-51</TD><TD >-1+(-51)=-52</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-17</TD><TD >-3+(-17)=-20</TD></tr>\n" );
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document.write( "From this list we can see that -3 and -17 add up to -20 and multiply to 51\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-20b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-20b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20b ALIGN=MIDDLE  ALT=\"-20b\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3b%2B-17b ALIGN=MIDDLE  ALT=\"-3b%2B-17b\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3b%2B-17b ALIGN=MIDDLE  ALT=\"-3b%2B-17b\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20b ALIGN=MIDDLE  ALT=\"-20b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20b ALIGN=MIDDLE  ALT=\"-20b\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Bhighlight%28-3b%2B-17b%29%2B51 ALIGN=MIDDLE  ALT=\"b%5E2%2Bhighlight%28-3b%2B-17b%29%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-3b-17b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-3b-17b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2-3b%29%2B%28-17b%2B51%29 ALIGN=MIDDLE  ALT=\"%28b%5E2-3b%29%2B%28-17b%2B51%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28b-3%29-17%28b-3%29 ALIGN=MIDDLE  ALT=\"b%28b-3%29-17%28b-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17 ALIGN=MIDDLE  ALT=\"-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-17%29%28b-3%29 ALIGN=MIDDLE  ALT=\"%28b-17%29%28b-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b-3 ALIGN=MIDDLE  ALT=\"b-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-3b-17b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-3b-17b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-17%29%28b-3%29 ALIGN=MIDDLE  ALT=\"%28b-17%29%28b-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-20b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-20b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-17%29%28b-3%29 ALIGN=MIDDLE  ALT=\"%28b-17%29%28b-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-20b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-20b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-3b-17b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-3b-17b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-20b%2B51 ALIGN=MIDDLE  ALT=\"b%5E2-20b%2B51\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-17%29%28b-3%29 ALIGN=MIDDLE  ALT=\"%28b-17%29%28b-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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