document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=906173>Question 906173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x to the fourth power - 20 x squared + 64 is greater than or equal to 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #549678 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=549678\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your problem is:\r<BR>\n" );
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document.write( "x^4 - 20x^2 + 64 >= 0\r<BR>\n" );
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document.write( "solve for x^4 - 20x^2 + 64 = 0 to find out when the graph crosses the x-axis.\r<BR>\n" );
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document.write( "this is a quadratic equation in disguise.\r<BR>\n" );
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document.write( "let y = x^2 and the equation becomes y^2 - 20y + 64 = 0\r<BR>\n" );
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document.write( "factor this equation to get (y-4)*(y-16)=0<BR>\n" );
document.write( "solve for y to get y = 4 and y = 16\r<BR>\n" );
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document.write( "since y = x^2 and you want to solve for x, now replace y with x^2 to get:\r<BR>\n" );
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document.write( "x^2 = 4 and x^2 = 16\r<BR>\n" );
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document.write( "solve for x to get x = plus or minus 2 and x = plus or minus 4\r<BR>\n" );
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document.write( "if you did this correctly, your graph crosses the x-axis at x = -2, +2, -4, +4\r<BR>\n" );
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document.write( "since your original equation is x^2-20x+64 >= 0, then you need to test each interval of the graph to see where the graph is greater than 0 and less than 0.\r<BR>\n" );
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document.write( "since your zero points are -2, +2, -4, +4, then the intervals you need to test are:\r<BR>\n" );
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document.write( "x < -4<BR>\n" );
document.write( "-4 < x < -2<BR>\n" );
document.write( "-2 < x < 2<BR>\n" );
document.write( "2 < x < 4<BR>\n" );
document.write( "x > 4\r<BR>\n" );
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document.write( "you test by picking a point in each interval and testing if the graph is positive or negative at that point.\r<BR>\n" );
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document.write( "for example, at the point x = -3, the graph of x^4 - 20x^2 + 64 is equal to (-3)^4 - 20(-3)^2 + 64 = -35 which is negative.\r<BR>\n" );
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document.write( "this means that the graph is negative in the interval -4 < x < -2\r<BR>\n" );
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document.write( "you do that at all the other intervals and you'll get a picture of when the graph is above the x-axis and below the x-axis.   \r<BR>\n" );
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document.write( "since the equation says that x^4 - 20x + 64 >= 0, then you pick the intervals that make that equation true.\r<BR>\n" );
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document.write( "as it turns out:\r<BR>\n" );
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document.write( "the intervals of x that satisfy the equation of x^2 - 20x + 64 >= 0 are:\r<BR>\n" );
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document.write( "x <= -4<BR>\n" );
document.write( "-2 <= x <= 2<BR>\n" );
document.write( "x >= 4\r<BR>\n" );
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document.write( "the graph of your equation is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://theo.x10hosting.com/2014/092602.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "graphing is simpler, but if you didn't have graphing capability, then you needed to test each interval to find out if the equation was greater than or equal to 0 in that interval.\r<BR>\n" );
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