document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250595>Question 250595</A>: <I><SPAN STYLE=\"word-break: break-all;\"> factoring   completely\r<BR>\n" );
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document.write( "16x^2 + 24x + 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182487 by </B><A HREF=/tutors/aboutme.mpl?userid=PRMath><B>PRMath(133)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=PRMath><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=182487\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+24X+%2B+9 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+24X+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To factor this, you sort of do a \"reverse\" of FOIL.  You know that FOIL stands for First, Outside, Inside, Last.  \r<BR>\n" );
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document.write( "Usually with FOIL, we start with two sets of parenthesis:\r<BR>\n" );
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document.write( "We want the first term in each set of parenthesis to multiply together to reach <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So, let's think of a fairly typical way that you can reach 16 when you multiply two numbers together.  How about the number 4?  \r<BR>\n" );
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document.write( "4 times 4 = 16 yes?  We cannot forget there is an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term there, so let's put in that x times x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "We now have:\r<BR>\n" );
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document.write( "(4X ________)(4X ________)\r<BR>\n" );
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document.write( "Now in the equation you gave: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+24X+%2B+9 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+24X+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >  we have to come up with a way that will let us MULTIPLY two numbers to reach 9 and yet ADD together to reach 24.   \r<BR>\n" );
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document.write( "A pretty typical way to reach 9 thru multiplication of two numbers is to use 3 times 3.\r<BR>\n" );
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document.write( "So, now we have:\r<BR>\n" );
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document.write( "(4X ?????  3)(4x  ?????   3).\r<BR>\n" );
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document.write( "What do we do with the 3?  Is it positive or negative?  Well the 9 in the equation is positive, as is the 24, so let's try a positive 3.......\r<BR>\n" );
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document.write( "(4x + 3)(4x + 3)\r<BR>\n" );
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document.write( "Does this give us our original equation of:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+24X+%2B+9 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+24X+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "Let's check:\r<BR>\n" );
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document.write( "FOIL:\r<BR>\n" );
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document.write( "First terms:  4x times 4x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Outside terms:  4x times 3 = 12x<BR>\n" );
document.write( "Inside terms:   4x times 3 = 12x<BR>\n" );
document.write( "Last terms:  3 times 3 = 9\r<BR>\n" );
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document.write( "That means we have:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + 12x + 12x + 9\r<BR>\n" );
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document.write( "Let's combine like terms of 12x + 12x = 24X and rewrite what we have:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + 24x + 9  \r<BR>\n" );
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document.write( "Ok! That's our original equation.  Therefore, (4x + 3)(4x + 3) \r<BR>\n" );
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document.write( "(or you can write it as:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x+%2B+3%29%5E2 ALIGN=MIDDLE  ALT=\"%284x+%2B+3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is a way to factor:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + 24x = 9  \r<BR>\n" );
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document.write( "I hope this helps. :-) </SPAN>\n" );
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