document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=259039>Question 259039</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a)What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ?<BR>\n" );
document.write( "b)Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots.<BR>\n" );
document.write( "c)What is the nature of the roots of 2x^2 - x - 3 = -5?\r<BR>\n" );
document.write( "\n" );
document.write( "I asked these together because they are all similar. Thanks for any help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #190729 by </B><A HREF=/tutors/aboutme.mpl?userid=drk><B>drk(1908)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drk><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=190729\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Lets take one at a time:<BR>\n" );
document.write( "a) What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ?<BR>\n" );
document.write( "x = 3 is a root means that (x-3)=0.<BR>\n" );
document.write( "Now, (x-3)(2x+1) = 2x^2 -5x -3, so k = 5.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "b) Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots.<BR>\n" );
document.write( "If a quadratic has two equal roots the discriminant, or D = b^2-4ac = 0.<BR>\n" );
document.write( "Applying this to our function, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+%28-4%29%5E2+-+4%2Ak%2Ak ALIGN=MIDDLE  ALT=\"d+=+%28-4%29%5E2+-+4%2Ak%2Ak\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "16-4k^2 = 0<BR>\n" );
document.write( "16 = 4k^2<BR>\n" );
document.write( "4 = k^2<BR>\n" );
document.write( "K = +-2<BR>\n" );
document.write( "--<BR>\n" );
document.write( "c) What is the nature of the roots of 2x^2 - x - 3 = -5? <BR>\n" );
document.write( "When they say nature of the roots, we are going after the quadratic formula which is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "based on our polynomial, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A2%2A%28-3%29+%29%29%2F%282%2A2%29+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A2%2A%28-3%29+%29%29%2F%282%2A2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-+sqrt%28+25%29%29%2F%284%29+ ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-+sqrt%28+25%29%29%2F%284%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B-5%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%281+%2B-5%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we have two real rational roots. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );