document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=78833>Question 78833</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely.    <BR>\n" );
document.write( "3x^2 – 23x + 40\r<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 #56571 by </B><A HREF=/tutors/aboutme.mpl?userid=mathdoc314><B>mathdoc314(58)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathdoc314><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=56571\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is fun to factor completely\r<BR>\n" );
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document.write( "It is going to be a product of two linear terms\r<BR>\n" );
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document.write( "(ax + b) (cx + d)\r<BR>\n" );
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document.write( "For a and c I think it is going to be 1 and 3, but not sure which is which\r<BR>\n" );
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document.write( "For b and d it has to multiply to 40 so the choices are<BR>\n" );
document.write( "1*40, 2*20, 4*10, or 5*8 \r<BR>\n" );
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document.write( "Since the middle term is a minus I think b and d are going to be negative, or you can write it\r<BR>\n" );
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document.write( "(x - b) (3x - d)\r<BR>\n" );
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document.write( "How will it work out so the middle term is -23\r<BR>\n" );
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document.write( "3b + d = 23<BR>\n" );
document.write( "I bet it is the 5 and 8 one<BR>\n" );
document.write( "3*5 + 8 = 23\r<BR>\n" );
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document.write( "(x - 5)(3x - 8) = 3x^2 -15x -8x + 40<BR>\n" );
document.write( "3x^2 - 23x + 40 = 0\r<BR>\n" );
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document.write( "That is how I learned to do it before I knew Completing the Square and the Quadratic Formula.\r<BR>\n" );
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document.write( "Completing the Square is a clearer process but you have to fool around with fractions\r<BR>\n" );
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document.write( "3x^2 - 23x + 40 = 0<BR>\n" );
document.write( "x^2 -23/3 x + 40/3 = 0\r<BR>\n" );
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document.write( "On the left I want to see something that looks like x^2 + 2Cx + C^2, so I determine that C is -23/6.  I add (-23/6)^2 to both sides to have this come out right.  I move the 40/3 to the right so I can solve for the squared term on the left.\r<BR>\n" );
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document.write( "x^2 + 2(-23/6) x + (-23/6)^2 = -40/3 + (-23/6)^2  \r<BR>\n" );
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document.write( "There I added the term I will need, (-23/6)^2, to both sides<BR>\n" );
document.write( "in order to have the left side be a perfect square.  Notice<BR>\n" );
document.write( "how the left side factors now:\r<BR>\n" );
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document.write( "(x - 23/6)^2 = -40/3 + 529/36 = (529 - 480)/36 = (49/36)\r<BR>\n" );
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document.write( "x - 23/6 = 7/6 or -7/6<BR>\n" );
document.write( "x = (23+7)/6 = 5<BR>\n" );
document.write( "or x = (23 - 7)/6 = 16/6\r<BR>\n" );
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document.write( "Therefore again the answer is 3(x-5)(x-8/3) or (x-5)(3x-8).\r<BR>\n" );
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