document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=63379>Question 63379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for x by completing the square:\r<BR>\n" );
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document.write( "kx^2-3cx-4a=0\r<BR>\n" );
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document.write( "I have tried but I wonder if there is no solution.  Thanks for your time!<BR>\n" );
document.write( "By the way, this question is not from any textbook.  My teacher had it on the board for the class to solve. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #44110 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=44110\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We will solve for x by completing the square:\r<BR>\n" );
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document.write( "kx^2-3cx-4a=0 First, divide both sides by k(assuming k is nonzero). We get:<BR>\n" );
document.write( "x^2-(3c/k)x-4(a/k)=0 Next, add 4(a/k) to both sides and we have: <BR>\n" );
document.write( "x^2-(3c/k)x=4(a/k) Now we add (9c^2/4k^2) to both sides. Why?  Because we want to make the left side of the equation a perfect square. (9c^2/4k^2)was selected because the sum of the factors (-3c/2k)^2  will yield the second term.\r<BR>\n" );
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document.write( "x^2-(3c/k)x+(9c^2/4k^2)=4(a/k)+(9c^2/4k^2) Now we factor the left side and get:<BR>\n" );
document.write( "(x-3c/2k)^2=4(a/k)+(9c^2/4k^2)  Now, we take the sqrt of both sides and we have\r<BR>\n" );
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document.write( "x-3c/2k=+or-sqrt(4(a/k)+(9c^2/4k^2)) simplifying the left side and adding (3c/2k) to both sides, we get<BR>\n" );
document.write( "x=3c/2k+or-sqrt((16ak+9c^2)/4k^2) further simplifying, we have<BR>\n" );
document.write( "x=(3c+or-sqrt(16ak+9c^2))/2k\r<BR>\n" );
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document.write( "You can re-check the work but this is my approach to solving quadratics by completing the square.\r<BR>\n" );
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document.write( "When factoring quadratics, it useful to remember that when the first term is one, the sum of the factors of the third term will equal the second term.<BR>\n" );
document.write( "Example: Find the factors of x^2-6x+9.  By inspection, we see that 3^2 is 9 and 3+3is 6 so the factors are (x-3)(x-3)\r<BR>\n" );
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document.write( "Hope this helps and have a nice holiday season---ptaylor<BR>\n" );
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