document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109304>Question 1109304</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For how many integers x is the number x^4 - 51x^2 + 50 negative? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724295 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=724295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the equation is negative between -sqrt(50) and -1, and between 1 and sqrt(50).\r<BR>\n" );
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document.write( "sqrt(50) is equal to 7.071 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "therefore, the equation is negative between -7.071 and -1, and between 1 and 7.071.\r<BR>\n" );
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document.write( "it is 0 at -7.071 and 0 at -1 and 0 at 1 and 0 at 7.071, therefore it is not negative at those values of x.\r<BR>\n" );
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document.write( "it is negative from -7 to -2, and from 2 to 7, inclusive.\r<BR>\n" );
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document.write( "that would be 12 integers total.\r<BR>\n" );
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document.write( "you find the 0 points by factoring the equation.\r<BR>\n" );
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document.write( "this is a fourth degree equation that can be solved as if it is a quadratic equation.\r<BR>\n" );
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document.write( "the equation is x^4 -51x^2 + 50 = 0 in standard form.\r<BR>\n" );
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document.write( "let z = x^2 and it becomes z^2 - 51z + 50 = 0\r<BR>\n" );
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document.write( "factor this quadratic equation to get (z-50) * (z-1) = 0\r<BR>\n" );
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document.write( "solve for z to get z = 50 or z = 1\r<BR>\n" );
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document.write( "since z = x^2, you get x^2 = 50 or x^2 = 1\r<BR>\n" );
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document.write( "solve for x to get x = plus or minus sqrt(50) or x = plus or minus 1.\r<BR>\n" );
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document.write( "those are the zero crossing points of x^4 - 51x^2<BR>\n" );
document.write( " + 50\r<BR>\n" );
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document.write( "now you check the intervals between all these zero crossing points.\r<BR>\n" );
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document.write( "you will find that the graph starts off positive until it gets to -sqrt(50), then goes negative until it gets to -1, then goes positive until it gets to 1, then goes negative until it gets to sqrt(50), then goes positive the rest of the way.\r<BR>\n" );
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document.write( "the graph of the equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/021208.jpg\" alt=\"$$$\">\r<BR>\n" );
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document.write( "from the graph, you can see that the equation is negative between -7.071 and -1, and it is negative between 1 and 7.071.\r<BR>\n" );
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document.write( "7.071 is the square root of 50 rounded to 3 decimal places.\r<BR>\n" );
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document.write( "there are 6 integers between -7.071 and -1, not including -7.071 and -1.<BR>\n" );
document.write( "they are -7, -6, -5, -4, -3, -2.\r<BR>\n" );
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document.write( "there are 6 integers between 1 and 7.071, not including 1 and 7.071.<BR>\n" );
document.write( "they are 2, 3, 4, 5, 6, 7.\r<BR>\n" );
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document.write( "that's 12 integers total.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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