document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=5357>Question 5357</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having trouble understanding how to use the Perfect Square Trinomial Formula.  my problem is:\r<BR>\n" );
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document.write( "9(x-3)^2-24(x-3)+16\r<BR>\n" );
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document.write( "I really don't even know where to begin.  I have the formula:\r<BR>\n" );
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document.write( "a^2+2ab+b^2=(a+b)^2<BR>\n" );
document.write( "a^2-2ab+b^2=(a-b)^2\r<BR>\n" );
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document.write( "Where do I go from here? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #2731 by </B><A HREF=/tutors/aboutme.mpl?userid=longjonsilver><B>longjonsilver(2297)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=longjonsilver><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/cgi-bin/show-face.mpl?user=longjonsilver><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=2731\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if you do not understand factorisation, then attempting to factorise 9(x-3)^2-24(x-3)+16 is pointless for you, as it is several steps up from \"basic\".\r<BR>\n" );
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document.write( "The 2 \"formulae\" you quoted are a start, but you need to understand exactly what you are trying to do, so here goes...\r<BR>\n" );
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document.write( "y= x^2 +6x + 8 is a quadratic, ie it is a u-shaped curve. Now, this u-shaped curve may well cross the x-axis when you plot it. The point(s) that any curve crosses the x-axis are called the solutions or roots of the equations and it is these points that a lot of algebra is concerned with.\r<BR>\n" );
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document.write( "How do we find where the equation crosses the x-axis? Well the x-axis is the line y=0, since every point on the x-axis has y equal to zero. So, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we want to know the x-values when y=0, ie we want to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, writing the equation this way doesn't help us. However, writing the equation as (one thing)*(another thing) is VERY helpful, since it equals zero remember: we now have 2 \"things\" multiplied that equal zero.\r<BR>\n" );
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document.write( "2x3? ...nope --> that is 6<BR>\n" );
document.write( "-3x4? ...nope --> that is -12<BR>\n" );
document.write( "-1x-7? ...nope --> that is +7\r<BR>\n" );
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document.write( "how about 0x56? ...YES --> that is 0. In other words, 2 things multiplied together...one of them MUST be zero. Which one? either could be.\r<BR>\n" );
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document.write( "So that is why we factorise: ie convert something to 2 things multiplied together.\r<BR>\n" );
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document.write( "So... factorise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, well a general \"formula\" is:\r<BR>\n" );
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document.write( "(x+a)(x+b) = x^2 + (a+b)x + ab\r<BR>\n" );
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document.write( "So, we need 2 numbers such that they multiply to give the constant value (ab) and add together to give the coefficient of the x term.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> we need 2 numbers that multiply to give +8 and also add to give +6.\r<BR>\n" );
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document.write( "Start off by writing down all the factors of 8 you know:\r<BR>\n" );
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document.write( "1x8<BR>\n" );
document.write( "2x4\r<BR>\n" );
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document.write( "that is it..which of these pairs add to give 6? 2 and 4 Voila!\r<BR>\n" );
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document.write( "(x+2)(x+4) is the factorised version of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Check this by multiplying out the 2 brackets...\r<BR>\n" );
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document.write( "Anyway, we have (x+2)(x+4) = 0, so we then know that x+2=0 OR x+4=0.\r<BR>\n" );
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document.write( "So, x=-2 or x=-4...the 2 answers we need.\r<BR>\n" );
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document.write( "Right, now back to your question... \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28x-3%29%5E2-24%28x-3%29%2B16 ALIGN=MIDDLE  ALT=\"9%28x-3%29%5E2-24%28x-3%29%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "lets just write y instead of x-3...to make it easier to see: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9y%5E2-24y%2B16 ALIGN=MIDDLE  ALT=\"9y%5E2-24y%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Also, I shall write the 9 and 16 as squares: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y%29%5E2-24y%2B4%5E2 ALIGN=MIDDLE  ALT=\"%283y%29%5E2-24y%2B4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now can you see from your <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-2ab%2Bb%5E2=%28a-b%29%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-2ab%2Bb%5E2=%28a-b%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > formula that what we have now is VERY similar?... a is 3y and b is 4 (and 2ab does indeed equal 24y). So, we can factorise the equation to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y-4%29%5E2 ALIGN=MIDDLE  ALT=\"%283y-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ie (3y-4)(3y-4).\r<BR>\n" );
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document.write( "Now, we just put the x back in... <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%28x-3%29-4%29%5E2 ALIGN=MIDDLE  ALT=\"%283%28x-3%29-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is the answer. Check by multiplying out the 2 brackets that you get back to the original question.\r<BR>\n" );
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document.write( "Hope this has helped?\r<BR>\n" );
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document.write( "jon.<BR>\n" );
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