document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022596>Question 1022596</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let K be a real number, and consider the quadratic equation (k+1)x^2+4kx+2=0<BR>\n" );
document.write( "a. Show that the discriminant of (k+1)x^2+4kx+2=0 defines a quadratic formula of k.<BR>\n" );
document.write( "b. Find the zeros of the function in part (a), and make a sketch of its graph (NOTE: this is optional, I can do this by myself.)<BR>\n" );
document.write( "c. For what value of k are there two distinct real solutions to the original quadratic equation?<BR>\n" );
document.write( "d. For what value of k are there two complex solutions to the given quadratic equation?<BR>\n" );
document.write( "e. For what value of k is there only one solution to the given quadratic equation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #638315 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=638315\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For the function above, a = k+1, b = 4k, and c = 2\r<BR>\n" );
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document.write( "a.  The discriminant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac+=+%284k%29%5E2+-+4%28k%2B1%292+=+16k%5E2+-+8k+-+8 ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac+=+%284k%29%5E2+-+4%28k%2B1%292+=+16k%5E2+-+8k+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "c.  There will be two distinct real roots if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2+-+8k+-+8+%3E+0 ALIGN=MIDDLE  ALT=\"16k%5E2+-+8k+-+8+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Solving this inequality gives a solution of (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >,-1/2)u(1,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "d.  There will be two complex roots (conjugates of each other) if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2+-+8k+-+8+%3C+0 ALIGN=MIDDLE  ALT=\"16k%5E2+-+8k+-+8+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The solution will be the open interval (-1/2, 1).\r<BR>\n" );
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document.write( "e.  There will be a unique solution if the discriminant is EQUAL to zero.  Hence k = -1/2 or 1. </SPAN>\n" );
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