document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=139520>Question 139520</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What do I solve this by factoring?\r<BR>\n" );
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document.write( "49x^2 - 14x - 3\r<BR>\n" );
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document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101719 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=101719\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49x%5E2-14x-3 ALIGN=MIDDLE  ALT=\"49x%5E2-14x-3\"   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=49x%5E2 ALIGN=MIDDLE  ALT=\"49x%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=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 49 and -3 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 49 and the last coefficient -3 to get -147. Now what two numbers multiply to -147 and add to the  middle coefficient -14? Let's list all of the factors of -147:\r<BR>\n" );
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document.write( "Factors of -147:\r<BR>\n" );
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document.write( "1,3,7,21,49,147\r<BR>\n" );
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document.write( "-1,-3,-7,-21,-49,-147 ...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 -147\r<BR>\n" );
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document.write( "(1)*(-147)\r<BR>\n" );
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document.write( "(3)*(-49)\r<BR>\n" );
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document.write( "(7)*(-21)\r<BR>\n" );
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document.write( "(-1)*(147)\r<BR>\n" );
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document.write( "(-3)*(49)\r<BR>\n" );
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document.write( "(-7)*(21)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14\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\">-147</TD><TD >1+(-147)=-146</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-49</TD><TD >3+(-49)=-46</TD></tr><tr><TD  align=\"center\">7</TD><TD  align=\"center\">-21</TD><TD >7+(-21)=-14</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">147</TD><TD >-1+147=146</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">49</TD><TD >-3+49=46</TD></tr><tr><TD  align=\"center\">-7</TD><TD  align=\"center\">21</TD><TD >-7+21=14</TD></tr>\n" );
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document.write( "From this list we can see that 7 and -21 add up to -14 and multiply to -147\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=49x%5E2-14x-3 ALIGN=MIDDLE  ALT=\"49x%5E2-14x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14x ALIGN=MIDDLE  ALT=\"-14x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B-21x ALIGN=MIDDLE  ALT=\"7x%2B-21x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B-21x ALIGN=MIDDLE  ALT=\"7x%2B-21x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14x ALIGN=MIDDLE  ALT=\"-14x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14x ALIGN=MIDDLE  ALT=\"-14x\"   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=49x%5E2%2Bhighlight%287x%2B-21x%29%2B-3 ALIGN=MIDDLE  ALT=\"49x%5E2%2Bhighlight%287x%2B-21x%29%2B-3\"   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=49x%5E2%2B7x-21x-3 ALIGN=MIDDLE  ALT=\"49x%5E2%2B7x-21x-3\"   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=%2849x%5E2%2B7x%29%2B%28-21x-3%29 ALIGN=MIDDLE  ALT=\"%2849x%5E2%2B7x%29%2B%28-21x-3%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=7x%287x%2B1%29-3%287x%2B1%29 ALIGN=MIDDLE  ALT=\"7x%287x%2B1%29-3%287x%2B1%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=7x ALIGN=MIDDLE  ALT=\"7x\"   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=-3 ALIGN=MIDDLE  ALT=\"-3\"   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=%287x-3%29%287x%2B1%29 ALIGN=MIDDLE  ALT=\"%287x-3%29%287x%2B1%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=7x%2B1 ALIGN=MIDDLE  ALT=\"7x%2B1\"   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=49x%5E2%2B7x-21x-3 ALIGN=MIDDLE  ALT=\"49x%5E2%2B7x-21x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287x-3%29%287x%2B1%29 ALIGN=MIDDLE  ALT=\"%287x-3%29%287x%2B1%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=49x%5E2-14x-3 ALIGN=MIDDLE  ALT=\"49x%5E2-14x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287x-3%29%287x%2B1%29 ALIGN=MIDDLE  ALT=\"%287x-3%29%287x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49x%5E2-14x-3 ALIGN=MIDDLE  ALT=\"49x%5E2-14x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49x%5E2%2B7x-21x-3 ALIGN=MIDDLE  ALT=\"49x%5E2%2B7x-21x-3\"   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=%287x-3%29%287x%2B1%29=0 ALIGN=MIDDLE  ALT=\"%287x-3%29%287x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the factorization equal to zero\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x-3=0 ALIGN=MIDDLE  ALT=\"7x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%2B1=0 ALIGN=MIDDLE  ALT=\"7x%2B1=0\"   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=x=3%2F7 ALIGN=MIDDLE  ALT=\"x=3%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F7 ALIGN=MIDDLE  ALT=\"x=-1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "-------------------------\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So our solutions are\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2F7 ALIGN=MIDDLE  ALT=\"x=3%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F7 ALIGN=MIDDLE  ALT=\"x=-1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
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