document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102632>Question 1102632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value of k in x^4-4x^2+k that gives 4 complex zeros. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717344 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=717344\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> in order for the equation to give you 4 complex zeros, the equaiton should not cross or touch the x-axis.\r<BR>\n" );
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document.write( "set the equation equal to 0 to get:\r<BR>\n" );
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document.write( "x^4 - 4x^2 + k = 0\r<BR>\n" );
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document.write( "let y = x^2.\r<BR>\n" );
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document.write( "the equation becomes y^2 - 4y + k = 0\r<BR>\n" );
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document.write( "this is a quadratic equation that can be solved through various means.\r<BR>\n" );
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document.write( "i'm not sure if you have to do it this way, but the use of the quadratic formula should be helpful.\r<BR>\n" );
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document.write( "in fact, if the discriminant is negative, then you will have complex zeroes.\r<BR>\n" );
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document.write( "now that you have the formula in standard form of ay^2 + by + c = 0, you can use the quadratic formula as shown below:\r<BR>\n" );
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document.write( "your formula is y^2 - 4y + k = 0<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( "use of the quadratic formula will get you:\r<BR>\n" );
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document.write( "y = -b + sqrt(b^2 - 4ac) / (2a) or y = -b - sqrt(b^2 - 4ac) / (2a)\r<BR>\n" );
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document.write( "b^2 - 4ac is the discriminant.\r<BR>\n" );
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document.write( "when b = -4 and a = 1 and c = k, you get:\r<BR>\n" );
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document.write( "b^2 - 4ax = (-4)^2 - 4*1*k)\r<BR>\n" );
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document.write( "this simplifies to 16 - 4k\r<BR>\n" );
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document.write( "is 16 - 4k is negative, then the solution will be complex.\r<BR>\n" );
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document.write( "set 16 - 4k < 0\r<BR>\n" );
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document.write( "add 4k to both sides of this equation to get 16 < 4k\r<BR>\n" );
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document.write( "divide both sides of this equation by 4 to get 4 < k\r<BR>\n" );
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document.write( "this means that k > 4\r<BR>\n" );
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document.write( "any value of k > 4 should do the trick.\r<BR>\n" );
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document.write( "we'll use 5 as a test.\r<BR>\n" );
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document.write( "when k = 5, your equation becomes y^2 - 4y + 5 = 0\r<BR>\n" );
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document.write( "solve this using quadratic formula to get:\r<BR>\n" );
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document.write( "y = 2+i or 2-i\r<BR>\n" );
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document.write( "since you have previously set y = x^2, then you get:\r<BR>\n" );
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document.write( "x^2 = 2+i or x^2 = 2-i\r<BR>\n" );
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document.write( "this leads to x = plus or minus sqrt(2+i) or x = plus or minus sqrt(2-i).\r<BR>\n" );
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document.write( "that's 4 complex roots of the equation x^4 - 4x^2 + 5 = 0\r<BR>\n" );
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document.write( "your factors will be (x-sqrt(2+i)) * (x+sqrt(2+i)) * (x-sqrt(2-i)) * (x+sqrt(2-i))\r<BR>\n" );
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document.write( "if you multiply those factors together, you will get x^4 - 4x^2 + 5.\r<BR>\n" );
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document.write( "if you graph x^4 - 4x^2 + 5, you will see that it doesn't cross the x-axis.\r<BR>\n" );
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document.write( "since it is a degree 4 equation, then it must have 4 roots, therefore all the roots have to be complex.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/112603.jpg\"alt=\"$$$\">\r<BR>\n" );
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