document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=214986>Question 214986</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with sloving quadractic equations by factoring:<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #162443 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=162443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, we need to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2%2B13x%2B7 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7\"   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=-2x%5E2%2B13x%2B7 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%282x%5E2-13x-7%29 ALIGN=MIDDLE  ALT=\"-%282x%5E2-13x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now let's try to factor the inner expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-13x-7 ALIGN=MIDDLE  ALT=\"2x%5E2-13x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-13x-7 ALIGN=MIDDLE  ALT=\"2x%5E2-13x-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that the first coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second coefficient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7 ALIGN=MIDDLE  ALT=\"-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now multiply the first coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the last term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7 ALIGN=MIDDLE  ALT=\"-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%29%28-7%29=-14 ALIGN=MIDDLE  ALT=\"%282%29%28-7%29=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now the question is: what two whole numbers multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product) <font size=4><b>and</b></font> add to the second coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous product).\r<BR>\n" );
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document.write( "Factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "1,2,7,14\r<BR>\n" );
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document.write( "-1,-2,-7,-14\r<BR>\n" );
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document.write( "Note: list the negative of each factor. 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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "1*(-14) = -14<BR>\n" );
document.write( "2*(-7) = -14<BR>\n" );
document.write( "(-1)*(14) = -14<BR>\n" );
document.write( "(-2)*(7) = -14\r<BR>\n" );
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document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD   align=\"center\"><font color=red>1</font></TD><TD   align=\"center\"><font color=red>-14</font></TD><TD   align=\"center\"><font color=red>1+(-14)=-13</font></TD></tr><tr><TD   align=\"center\"><font color=black>2</font></TD><TD   align=\"center\"><font color=black>-7</font></TD><TD   align=\"center\"><font color=black>2+(-7)=-5</font></TD></tr><tr><TD   align=\"center\"><font color=black>-1</font></TD><TD   align=\"center\"><font color=black>14</font></TD><TD   align=\"center\"><font color=black>-1+14=13</font></TD></tr><tr><TD   align=\"center\"><font color=black>-2</font></TD><TD   align=\"center\"><font color=black>7</font></TD><TD   align=\"center\"><font color=black>-2+7=5</font></TD></tr>\n" );
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document.write( "From the table, we can see that the two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the middle coefficient).\r<BR>\n" );
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document.write( "So the two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > both multiply to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font size=4><b>and</b></font> add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now replace the middle term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13x ALIGN=MIDDLE  ALT=\"-13x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-14x ALIGN=MIDDLE  ALT=\"x-14x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-14 ALIGN=MIDDLE  ALT=\"-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13 ALIGN=MIDDLE  ALT=\"-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this shows us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-14x=-13x ALIGN=MIDDLE  ALT=\"x-14x=-13x\"   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=2x%5E2%2Bhighlight%28x-14x%29-7 ALIGN=MIDDLE  ALT=\"2x%5E2%2Bhighlight%28x-14x%29-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace the second term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13x ALIGN=MIDDLE  ALT=\"-13x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-14x ALIGN=MIDDLE  ALT=\"x-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=%282x%5E2%2Bx%29%2B%28-14x-7%29 ALIGN=MIDDLE  ALT=\"%282x%5E2%2Bx%29%2B%28-14x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the terms into two pairs.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%282x%2B1%29%2B%28-14x-7%29 ALIGN=MIDDLE  ALT=\"x%282x%2B1%29%2B%28-14x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the first group.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%282x%2B1%29-7%282x%2B1%29 ALIGN=MIDDLE  ALT=\"x%282x%2B1%29-7%282x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-7%29%282x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x-7%29%282x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. Or factor out the common term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B1 ALIGN=MIDDLE  ALT=\"2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%282x%5E2-13x-7%29 ALIGN=MIDDLE  ALT=\"-1%282x%5E2-13x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > then factors further to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-7%29%282x%2B1%29 ALIGN=MIDDLE  ALT=\"-%28x-7%29%282x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2%2B13x%2B7 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > completely factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-7%29%282x%2B1%29 ALIGN=MIDDLE  ALT=\"-%28x-7%29%282x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2%2B13x%2B7=-%28x-7%29%282x%2B1%29 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7=-%28x-7%29%282x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now let's use the factorization above to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2%2B13x%2B7=0 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7=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=-2x%5E2%2B13x%2B7=0 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B13x%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28x-7%29%282x%2B1%29=0 ALIGN=MIDDLE  ALT=\"-%28x-7%29%282x%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (see above)\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=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >    Now solve for x in each case\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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