document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203803>Question 203803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi again, I was wondering if someone could show me how to do the following;<BR>\n" );
document.write( "Factorize 6x^2 + x - 12.<BR>\n" );
document.write( "and Expand (x + 1)(-2x + 1)(x - 3).<BR>\n" );
document.write( "Both are seperate equations and I need different answers. If someone could show me how to do them that would be great.<BR>\n" );
document.write( "Thanks, -Nick </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153783 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=153783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi Nick....\r<BR>\n" );
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document.write( "Before we start... have you had much work with factoring?  I will give you a small example, just to be sure of what you understand, k?\r<BR>\n" );
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document.write( "For example... say you had:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2+%2B+8x+%2B+16 ALIGN=MIDDLE  ALT=\"X%5E2+%2B+8x+%2B+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >   What you have to say to yourself is:  What would we use to MULTIPLY to get to 16 and yet ADD to get to 8.  Well, a pretty typical way to get to 16 is to multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+%2A+2 ALIGN=MIDDLE  ALT=\"8+%2A+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > but... I can't see any way to add those up to get to 8.  Can you?  Nope, probably not.\r<BR>\n" );
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document.write( "How about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%2A+4 ALIGN=MIDDLE  ALT=\"4+%2A+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >?   That would multiply to be 16 and yet add to be 8.  So we'd put that info into an equation like this;\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+4%29%28x+%2B+4%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+4%29%28x+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  THEN if we use FOIL, we can see that we will indeed come up with our original equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2+%2B+8x+%2B+16 ALIGN=MIDDLE  ALT=\"X%5E2+%2B+8x+%2B+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  I'm assuming you know FOIL but if not, repost and I'll explain further.  \r<BR>\n" );
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document.write( "SO now that we have reviewed factoring a TINY bit, let's look at your original equation:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2+%2B+x+-+12 ALIGN=MIDDLE  ALT=\"6x%5E2+%2B+x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In this case, you have to get numbers that MULTIPLY to get to -12, YET they must ADD up to be +1.  There is a +1 in front of the \"x\" --- think of this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2+%2B+1x+-+12 ALIGN=MIDDLE  ALT=\"6x%5E2+%2B+1x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "BUT we have a 6 at the front of our equation.  That makes things harder because you have to choose first terms that multiply to be 6.  Sooo what you can do is sort of test some ideas..........\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286x+%2B+4%29%28x+-+3%29 ALIGN=MIDDLE  ALT=\"%286x+%2B+4%29%28x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  OH no.  That won't work at all.  We get -12 for multiplying but using FOIL, we get -18x + 4x.  That would give us -14x.  ARghhhhhhhhhh.  That's no good.\r<BR>\n" );
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document.write( "HOW about......<BR>\n" );
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document.write( "What can we multiply to get 12?  How about 3 times 4?  Let's plug that in.....\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x+%2B+3%29%283x+-+4%29 ALIGN=MIDDLE  ALT=\"%282x+%2B+3%29%283x+-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Could this be the factoring?  Let's check this with FOIL\r<BR>\n" );
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document.write( "First:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2A+3x+=+6x%5E2 ALIGN=MIDDLE  ALT=\"2x+%2A+3x+=+6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  GOOD<BR>\n" );
document.write( "Outside:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2A+-4+=+-8x ALIGN=MIDDLE  ALT=\"2x+%2A+-4+=+-8x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Ok.....<BR>\n" );
document.write( "Inside:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2A+3x+=+9x ALIGN=MIDDLE  ALT=\"3+%2A+3x+=+9x\"   DISABLED_style= \"max-width:90%; height:auto;\" >Ok.....<BR>\n" );
document.write( "Last:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2A+-4+=+-12 ALIGN=MIDDLE  ALT=\"3+%2A+-4+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >  GOOD.\r<BR>\n" );
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document.write( "So we ended up with:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2+-+8x+%2B+9x+-+12 ALIGN=MIDDLE  ALT=\"6x%5E2+-+8x+%2B+9x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Combine like terms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2+%2B+1x+-+12 ALIGN=MIDDLE  ALT=\"6x%5E2+%2B+1x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "OH so we have our equation.  Therefore we know that our factoring is correct. Yay.\r<BR>\n" );
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document.write( "Now to expand.................\r<BR>\n" );
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document.write( "(x + 1)(-2x + 1)(x - 3)\r<BR>\n" );
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document.write( "Let's just focus on the last two parts of the equation: (-2x + 1)(x - 3) <BR>\n" );
document.write( "Let's use FOIL on that:  \r<BR>\n" );
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document.write( "First:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x+%2A+x+=+-2x%5E2 ALIGN=MIDDLE  ALT=\"-2x+%2A+x+=+-2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Outside: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x+%2A+-3+=+6x ALIGN=MIDDLE  ALT=\"-2x+%2A+-3+=+6x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Inside:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2A+x+=+1x ALIGN=MIDDLE  ALT=\"1+%2A+x+=+1x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Last: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2A+-3+=+-3 ALIGN=MIDDLE  ALT=\"1+%2A+-3+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That gives us:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2+%2B+6x+%2B+1x+-+3 ALIGN=MIDDLE  ALT=\"-2x%5E2+%2B+6x+%2B+1x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Combine like terms:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2+%2B+7x+-+3 ALIGN=MIDDLE  ALT=\"-2x%5E2+%2B+7x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "BUT we are not done.  Don't forget we have another part of the equation...  We still have to multiply by (x + 1)\r<BR>\n" );
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document.write( "SO we mutiply like this:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+1%29%28-2x%5E2+%2B+7x+-+3%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+1%29%28-2x%5E2+%2B+7x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We take the \"x\" in (x + 1) and multiply it by EVERYTHING, like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2A+7x+=+7x%5E2 ALIGN=MIDDLE  ALT=\"x+%2A+7x+=+7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2A+-3+=+-3x ALIGN=MIDDLE  ALT=\"x+%2A+-3+=+-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sooo now we have:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E3+%2B+7x%5E2+-+3x ALIGN=MIDDLE  ALT=\"-2x%5E3+%2B+7x%5E2+-+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > BUT we are STILL not done...\r<BR>\n" );
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document.write( "We have to take the 1 in (x+ 1) and muiltiply it by EVERTHING, too! \r<BR>\n" );
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document.write( "Sooo now we have:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2+%2B+7x+-+3 ALIGN=MIDDLE  ALT=\"-2x%5E2+%2B+7x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We are STILL not done (and this would be SO much easier to show you in person.  Typing it all out is going to confuse you further, I'm afraid but here goes)...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E3+%2B+7x%5E2+-+3x+-2x%5E2+%2B+7x+-+3 ALIGN=MIDDLE  ALT=\"-2x%5E3+%2B+7x%5E2+-+3x+-2x%5E2+%2B+7x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine like terms......\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E3+%2B+5x%5E2+%2B+4x+-+3 ALIGN=MIDDLE  ALT=\"-2x%5E3+%2B+5x%5E2+%2B+4x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "THAT is your final answer.  Whewwwwwwww.\r<BR>\n" );
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document.write( "I hope this helps but my guess is that it will be very confusing.... sigh.\r<BR>\n" );
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