document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=959135>Question 959135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let f be the function defined by f(x)=ax^2- sqrt2 for some positive a. If f(f(sqrt2))=-sqrt2, find the exact value of a. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #586274 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=586274\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> looks like x = sqrt(2)/2 will do it.\r<BR>\n" );
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document.write( "f(x) is equal to ax^2 - sqrt(2)\r<BR>\n" );
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document.write( "what is the value of a when f(f(sqrt(2)) = -sqrt(2)?\r<BR>\n" );
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document.write( "if f(x) is equal to ax^2 - sqrt(2)), then f(sqrt(2)) is equal to a * sqrt(2)^2 - sqrt(2).\r<BR>\n" );
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document.write( "you are simply replacing x with sqrt(2).\r<BR>\n" );
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document.write( "f(sqrt(2)) is equal to a * sqrt(2)^2 - sqrt(2) which is equal to a * 2 - sqrt(2) which is equal to 2a - sqrt(2).\r<BR>\n" );
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document.write( "you have f(sqrt(2)) is equal to 2a - sqrt(2).\r<BR>\n" );
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document.write( "f(f(sqrt(2))) is equal to f(2a - sqrt(2)).\r<BR>\n" );
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document.write( "you are simply replacing f(sqrt(2)) with 2a - sqrt(2)).\r<BR>\n" );
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document.write( "f(2a - sqrt(2)) is equal to a * (2a - sqrt(2))^2 - sqrt(2)\r<BR>\n" );
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document.write( "you are simply replacing x in f(x) = a * x^2 - sqrt(2) with (2a - sqrt(2)).\r<BR>\n" );
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document.write( "since (2a - sqrt(2))^2 is equal to 4a^2 - 4a*sqrt(2) + 2, your equation of:\r<BR>\n" );
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document.write( "f(2a - sqrt(2)) is equal to a * (2a - sqrt(2))^2 - sqrt(2)becomes:\r<BR>\n" );
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document.write( "f(2a - sqrt(2)) is equal to a * (4a^2 - 4a*sqrt(2) + 2) - sqrt(2).\r<BR>\n" );
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document.write( "you know two things.<BR>\n" );
document.write( "1.  a has to be positive.<BR>\n" );
document.write( "2.  the end result has to be - sqrt(2).\r<BR>\n" );
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document.write( "this end result can only occur if a * (4a^2 - 4a*sqrt(2) + 2) is equal to 0.\r<BR>\n" );
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document.write( "since a has to be positive, this can only occur if 4a^2 - 4a*sqrt(2) + 2 is equal to 0.\r<BR>\n" );
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document.write( "since this is a quadratic equation, set it equal to 0 and factor it using the quadratic formula.\r<BR>\n" );
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document.write( "you will get a = sqrt(2)/2.\r<BR>\n" );
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document.write( "here's the graph of your final equation of y = a * (4a^2 - 4a*sqrt(2) + 2) - sqrt(2).\r<BR>\n" );
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document.write( "you can see that the graph intersect with the graph of the line y = -sqrt(2) in two places.\r<BR>\n" );
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document.write( "one place is when a = 0 but this is invalid because a has to be positive.\r<BR>\n" );
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document.write( "the other place is when a is equal to sqrt(2)/2\r<BR>\n" );
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document.write( "sqrt(2)/2 is equal to .7071 on the graph.\r<BR>\n" );
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document.write( "-sqrt(2) is equal to -1.414\r<BR>\n" );
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document.write( "the solution is at the point (x,y) = .7071,-1.414).\r<BR>\n" );
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