document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=168028>Question 168028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely by grouping\r<BR>\n" );
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document.write( "27x^2 - 117x - 90 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #123848 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=123848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 27x^2 - 117x - 90 factors out to be (3x-15)*(9x+6)<BR>\n" );
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document.write( "your c factor possibilities were:  +/- [ (90,1), (9,10), (6*15) ]<BR>\n" );
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document.write( "your a factor possibilities were:  +/- [ (27,1), (9,3) ]<BR>\n" );
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document.write( "since your factored equation will be in the form of (a1*x +c1) * (a2*x +c2), you need to find a combination that:<BR>\n" );
document.write( "a1*c2 + a2*c1 = -117<BR>\n" );
document.write( "c1*c2 = -90<BR>\n" );
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document.write( "i tried different combinations until i found the right one.<BR>\n" );
document.write( "let a1 = 3<BR>\n" );
document.write( "let a2 = 9<BR>\n" );
document.write( "let c1 = -15<BR>\n" );
document.write( "let c2 = 6<BR>\n" );
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document.write( "a1*c2 = 3*6 = 18<BR>\n" );
document.write( "a2*c1 = 9*(-15) = -135<BR>\n" );
document.write( "a1*c2 + a2*c1 = 18 + (-135) = 18 - 135 = -117<BR>\n" );
document.write( "c1*c2 = (-15)*6 = -90<BR>\n" );
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document.write( "before i even bothered, i used the part of the quadratic formula sqrt(b^2-4ac) to determine if i had integer roots or not.<BR>\n" );
document.write( "if the square root of b^2 - 4ac is an integer, then the quadratic equation has integer roots.<BR>\n" );
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document.write( "in the formula above,<BR>\n" );
document.write( "a = 27<BR>\n" );
document.write( "b = -117<BR>\n" );
document.write( "c = -90<BR>\n" );
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document.write( "b^2 = 13689<BR>\n" );
document.write( "4ac = -9720<BR>\n" );
document.write( "sqrt (b^2 - 4ac) = sqrt (13689 - (-9720) = sqrt (13689 + 9720 = sqrt (234098) = 153.<BR>\n" );
document.write( "the square root is an integer and so the quadratic equation has integer roots.<BR>\n" );
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document.write( "then i went looking for the roots as above.<BR>\n" );
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