document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182830>Question 1182830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The positive value of k which will make<BR>\n" );
document.write( "4x2 - 4kx + 4k + 5 = 0 a perfect square trinomial is _____.\r<BR>\n" );
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document.write( "-- Please help me.. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812929 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=812929\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "It appears to me that the response from the other tutor is only useful for a student who already knows how to solve the problem.<br><BR>\n" );
document.write( "Assuming you posted the question because you don't know how to solve the problem, I will provide a response that I hope will help you learn how.<br><BR>\n" );
document.write( "For the general quadratic equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "the roots are given by the quadratic formula,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The radicand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the discriminant -- it tells whether the equation has 0, 1, or 2 real zeros (roots).  If the discriminant is 0, then the equation has a single root; that means the quadratic is a perfect square trinomial.<br><BR>\n" );
document.write( "So you need the discriminant of the given trinomial to be 0.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%284%29x%5E2+%2B+%28-4k%29x+%2B+%284k%2B5%29 ALIGN=MIDDLE  ALT=\"y+=+%284%29x%5E2+%2B+%28-4k%29x+%2B+%284k%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this trinomial, a=4, b=-4k, and c=4k+5.  The discriminant is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+%28-4k%29%5E2-4%284%29%284k%2B5%29+=+16k%5E2-64k-80 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+%28-4k%29%5E2-4%284%29%284k%2B5%29+=+16k%5E2-64k-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the trinomial is a perfect square if the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2-64k-80 ALIGN=MIDDLE  ALT=\"16k%5E2-64k-80\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 0:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2-64k-80=0 ALIGN=MIDDLE  ALT=\"16k%5E2-64k-80=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2-4k-5=0 ALIGN=MIDDLE  ALT=\"k%5E2-4k-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k-5%29%28k%2B1%29=0 ALIGN=MIDDLE  ALT=\"%28k-5%29%28k%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5 ALIGN=MIDDLE  ALT=\"k=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: the trinomial is a perfect square if k is either 5 or -1.<br><BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "k=5: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-20x%2B25+=+%282x-5%29%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2-20x%2B25+=+%282x-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "k=-1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-4k%2B1+=+%282x-1%29%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2-4k%2B1+=+%282x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
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