document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=155144>Question 155144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many solutions exist for a quadratic equation?  How do we determine whether the solutions are real or complex? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #114222 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=114222\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "A quadratic polynomial can have 2 real solutions, 1 real solution, or 2 complex solutions. To determine whether the solutions are real or complex, simply use the discriminant formula. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For example, to see if the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B6x-10 ALIGN=MIDDLE  ALT=\"2x%5E2%2B6x-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > has real or complex solutions, let's use the discriminant formula\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B6x-10 ALIGN=MIDDLE  ALT=\"2x%5E2%2B6x-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-10 ALIGN=MIDDLE  ALT=\"c=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=b%5E2-4ac ALIGN=MIDDLE  ALT=\"D=b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the discriminant formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%286%29%5E2-4%282%29%28-10%29 ALIGN=MIDDLE  ALT=\"D=%286%29%5E2-4%282%29%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=2, b=6, and c=-10\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=36-4%282%29%28-10%29 ALIGN=MIDDLE  ALT=\"D=36-4%282%29%28-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=36--80 ALIGN=MIDDLE  ALT=\"D=36--80\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 4(2)(-10) to get (8)(-10)=-80\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=36%2B80 ALIGN=MIDDLE  ALT=\"D=36%2B80\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite D=36--80 as D=36+80\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=116 ALIGN=MIDDLE  ALT=\"D=116\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 36 to 80 to get 116\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the discriminant is greater than zero, this means that there are two real solutions. So the quadratic has two real solutions\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );