document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078340>Question 1078340</A>: <I><SPAN STYLE=\"word-break: break-all;\"> X^2-2kx-8k^2<BR>\n" );
document.write( "Find in terms of k, the least value for c for which the equation f(x)=c has a real solution for x.\r<BR>\n" );
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document.write( "Wth is this?<BR>\n" );
document.write( " Thanks for the help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #692851 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=692851\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = c = x^2-2kx-8k^2<BR>\n" );
document.write( "Subtract c from both sides:<BR>\n" );
document.write( "x^2-2kx-8k^2-c = 0<BR>\n" );
document.write( "This is a quadratic equation with coefficients 1, -2k and -8k^2 - c<BR>\n" );
document.write( "In order for there to be one real solution b^2 - 4ac = 0<BR>\n" );
document.write( "So we have (-2k)^2 - 4*(-8k^2 - c) = 0<BR>\n" );
document.write( "4k^2 + 32k^2 + 4c = 0<BR>\n" );
document.write( "9k^2 + c = 0<BR>\n" );
document.write( "Or c = -9k^2<BR>\n" );
document.write( "Thus c must be at least -9k^2 for one real solution<BR>\n" );
document.write( "With this value for c, the equation becomes x^2-2kx-8k^2 + 9k^2 = 0 -> x^2-2kx + k^2 = 0<BR>\n" );
document.write( "This can be factored as (x-k)(x-k) = 0, which gives one real solution, x = k. </SPAN>\n" );
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