document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=997637>Question 997637</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B8x%2Bk=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B8x%2Bk=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , for what values of k will the equation have a repeated root, 2 real roots, and 3 values of k that will give the equation rational roots. I understand what it means to have those types of roots, I just don't understand how you would get to those specific answers. Like, I know how to find and imaginary root for this equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%5B-infinite%3Ck%3Cless+than+0%5D+ ALIGN=MIDDLE  ALT=\"+%5B-infinite%3Ck%3Cless+than+0%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > . Would that be right? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #615643 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=615643\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 4x^2+8x+k=0<BR>\n" );
document.write( "4x^2+8x=-k<BR>\n" );
document.write( "Factor out 4.<BR>\n" );
document.write( "4(x^2+2x)= -k  but k can be any constant, so it doesn't have to be divided by 4.<BR>\n" );
document.write( "If you complete the square, the roots will be the same, -1.<BR>\n" );
document.write( "4(x^2+2x+1)= -(k)+4, adding 4 to both sides<BR>\n" );
document.write( "4(x+1)^2= -k +4<BR>\n" );
document.write( "The right side has to equal zero for completing the square to give two roots of -1<BR>\n" );
document.write( "-k=-4<BR>\n" );
document.write( "k=4<BR>\n" );
document.write( "4x^2+8x+4=0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B8x%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B8x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "========<BR>\n" );
document.write( "Two real roots can be many things, so long as the discriminant b^2-4ac is positive.<BR>\n" );
document.write( "b^2>4ak, since k is acting as c<BR>\n" );
document.write( "64>16k<BR>\n" );
document.write( "k<4 will work<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B8x%2B2%2C4x%5E2%2B8x-2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C4x%5E2%2B8x%2B2%2C4x%5E2%2B8x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=========<BR>\n" );
document.write( "Rational roots work if the discriminant gives a perfect square, for all parts of the quadratic formula will be a fraction with nothing but integers.<BR>\n" );
document.write( "b^2-16k has to be a perfect square<BR>\n" );
document.write( "64-16k\r<BR>\n" );
document.write( "\n" );
document.write( "k=0 works and roots are 4 and 0<BR>\n" );
document.write( "k=3 :(1/8){-8 +/- sqrt(16){=(1/8)(-4) and (1/8)(-12)<BR>\n" );
document.write( "k= -36/16 (1/8) (-8 +/- sqrt (100) and (1/8)(-8+10), (1/8) (-8-10) or (10/8), (-18/8)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );