document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1002043>Question 1002043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value/s of k so that the minimum value of f(x) = x^2 + kx + 3 is the same as the maximum value of g(x) = k + 4x - x^2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619073 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=619073\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your two equations are:\r<BR>\n" );
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document.write( "x^2 + kx + 3 and -x^2 + 4x + k\r<BR>\n" );
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document.write( "you want to find the max value of x^2 + kx + 3 and the min value of -x^2 + 4x + k and then set them equal to each other and then solve for k.\r<BR>\n" );
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document.write( "since these are in standard quadratic form, look for x = -b/2a in each.\r<BR>\n" );
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document.write( "x^2 + kx + 3 gets you:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = k<BR>\n" );
document.write( "c = 3\r<BR>\n" );
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document.write( "x = -b/2a = -k/2\r<BR>\n" );
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document.write( "-x^2 + 4x + k gets you:<BR>\n" );
document.write( "a = -1<BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "c = k\r<BR>\n" );
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document.write( "x = -b/2a gets you x = -4/-2 = 2\r<BR>\n" );
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document.write( "the min value of y for x^2 + kx + 3 will be at f(-b/2a) = f(-k/2).\r<BR>\n" );
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document.write( "the max value of y for -x^2 + 4x + k will be at f(-b/2a) = f(2).\r<BR>\n" );
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document.write( "f(-k/2) for x^2 + kx + 3 becomes (-k/2)^2 + k(-k/2) + 3.<BR>\n" );
document.write( "simplify this equation to get:<BR>\n" );
document.write( "f(-k/2) = -k^2/4 + 3\r<BR>\n" );
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document.write( "f(2) for -x^2 + 4x + k becomes -(2^2) + 4(2) + k.<BR>\n" );
document.write( "simplify this equation to get:<BR>\n" );
document.write( "f(2) = k + 4\r<BR>\n" );
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document.write( "f(-k/2) is the min value of x^2 + kx + 3<BR>\n" );
document.write( "f(2) is the max value of -x^2 + 4x + k\r<BR>\n" );
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document.write( "set them equal to each other and you get:<BR>\n" );
document.write( "f(-k/2) = f(2) which becomes:<BR>\n" );
document.write( "-k^2/4 + 3 = k + 4\r<BR>\n" );
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document.write( "add k^2/4 to both sides of the equation and subtract 3 from both sides of the equation to get:\r<BR>\n" );
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document.write( "0 = k^2/4 + k + 4 - 3<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "0 = k^2/4 + k + 1<BR>\n" );
document.write( "multiply both sides of this equation by 4 to get:<BR>\n" );
document.write( "0 = k^2 + 4k + 4\r<BR>\n" );
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document.write( "factor k^2 + 4k + 4 to get (k+2)*(k+2) = 0<BR>\n" );
document.write( "solve for k to get k = -2.\r<BR>\n" );
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document.write( "the min point of x^2 + kx + 3 and the max point of -x^2 + 4x + k should be the same when k = -2.\r<BR>\n" );
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document.write( "replace k with -2 in x^2 + kx + 3 to get x^2 -2x + 3.\r<BR>\n" );
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document.write( "replace k with -2 in -x^2 + 4x + k to get -x^2 + 4x - 2.\r<BR>\n" );
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document.write( "min point of x^2 - 2x +3 is at (-b/2a,f(-b/2a))<BR>\n" );
document.write( "-b/2a = 2/2 = 1<BR>\n" );
document.write( "f(-b/2a) = f(1) = 1 - 2 + 3 = 2.\r<BR>\n" );
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document.write( "max point of -x^2 + 4x - 2 is at (-b/2a,f(-b/2a))<BR>\n" );
document.write( "-b/2a = -4/-2 = 2<BR>\n" );
document.write( "f(2) = -4 + 8 - 2 = 2.\r<BR>\n" );
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document.write( "min point of x^2 - 2x + 3 is at (1,2)<BR>\n" );
document.write( "max point of -x^2 + 4x - 2 is at (2,2)\r<BR>\n" );
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document.write( "the y values are the same so the min value and max value are the same.\r<BR>\n" );
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document.write( "the graph of both equations is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/110801.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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