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" );
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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 #190728 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=190728\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)<BR>\n" );
document.write( "2x^2 - kx - 3=0<BR>\n" );
document.write( "just plug in 3 for x and solve for k<BR>\n" );
document.write( "2*3^2 -3k - 3=0<BR>\n" );
document.write( "2*9-3k-3=0<BR>\n" );
document.write( "15=3k<BR>\n" );
document.write( "5=k<BR>\n" );
document.write( "b)<BR>\n" );
document.write( "b^2-4ac must =0<BR>\n" );
document.write( "(-1)^2-4(k*k)=0<BR>\n" );
document.write( "1=4k^2<BR>\n" );
document.write( "1/4=k^2<BR>\n" );
document.write( "k=+1/2 and -1/2<BR>\n" );
document.write( "(-1)^2-4*(-1/2)(-1/2<BR>\n" );
document.write( "1-1=0<BR>\n" );
document.write( "also works with +1/2<BR>\n" );
document.write( "c)<BR>\n" );
document.write( "2x^2 - x - 3 = -5<BR>\n" );
document.write( "2x^2-x+2=0\r<BR>\n" );
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document.write( " two complex numbers because the discriminant(b^2-4ac=(-1)^2-(4*2*(-3)) is less than zero\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B-1x%2B2+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-1x%2B2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <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;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <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;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-1%29%5E2-4%2A2%2A2=-15 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-1%29%5E2-4%2A2%2A2=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -15 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -15 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+15%29+=+3.87298334620742 ALIGN=MIDDLE  ALT=\"sqrt%28+15%29+=+3.87298334620742\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28--1%2B-+i%2Asqrt%28+-15+%29%29%2F2%5C2+=++%28--1%2B-+i%2A3.87298334620742%29%2F2%5C2+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-1%2Ax%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-1%2Ax%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );