document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=707046>Question 707046</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the value(s) of \"k\" for which there is one and only one real solution to the following quadratic: 10kx = -16x^2 - 25\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #435526 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=435526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There will be one and only one real solution to a quadratic equation when the discriminant is zero. So let's start by putting the equation into standard form. Adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2 ALIGN=MIDDLE  ALT=\"16x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 25 to each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2%2B10kx%2B25=0 ALIGN=MIDDLE  ALT=\"16x%5E2%2B10kx%2B25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The discriminant, in general, is <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;\" >. Our \"a\" is 16, \"b\" is 10k and \"c\" is 25. So our discriminant is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810k%29%5E2-3%2816%29%2825%29 ALIGN=MIDDLE  ALT=\"%2810k%29%5E2-3%2816%29%2825%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100k%5E2-1200 ALIGN=MIDDLE  ALT=\"100k%5E2-1200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We want this to be zero so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100k%5E2-1200+=+0 ALIGN=MIDDLE  ALT=\"100k%5E2-1200+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we solve for k. Dividing both sides by 100:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2-12+=+0 ALIGN=MIDDLE  ALT=\"k%5E2-12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding 12:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2+=+12 ALIGN=MIDDLE  ALT=\"k%5E2+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+sqrt%2812%29 ALIGN=MIDDLE  ALT=\"k+=+sqrt%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-sqrt%2812%29 ALIGN=MIDDLE  ALT=\"k+=+-sqrt%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. These square roots simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+sqrt%284%2A3%29 ALIGN=MIDDLE  ALT=\"k+=+sqrt%284%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-sqrt%284%2A3%29 ALIGN=MIDDLE  ALT=\"k+=+-sqrt%284%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+sqrt%284%29%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"k+=+sqrt%284%29%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-sqrt%284%29%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"k+=+-sqrt%284%29%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"k+=+2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"k+=+-2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "These are the only possible values for k which will lead to one and only one real solution for x. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );