document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=838283>Question 838283</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My name is Robert Brown and the question is:<BR>\n" );
document.write( "\"Without solving, determine the character of the solutions of each equation in the complex number system. Verify your answer using a graphing utility.\"<BR>\n" );
document.write( "73) 3x^2 - 3x +4=0\r<BR>\n" );
document.write( "\n" );
document.write( "I have a test TODAY and no one at the school is willing to help me </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #504962 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=504962\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> SOLUTION (or verification) BY GRAPHING:<BR>\n" );
document.write( "That is a quadratic equation.<BR>\n" );
document.write( "The left hand side, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-3x%2B4 ALIGN=MIDDLE  ALT=\"3x%5E2-3x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a polynomial of degree 2 (a quadratic polynomial).<BR>\n" );
document.write( "The function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2-3x%2B4 ALIGN=MIDDLE  ALT=\"y=3x%5E2-3x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic function.<BR>\n" );
document.write( "Quadratic functions graph as parabolas.<BR>\n" );
document.write( "If you graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2-3x%2B4 ALIGN=MIDDLE  ALT=\"y=3x%5E2-3x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you can see where the graph crosses or touches the x-axis (if it ever does).<BR>\n" );
document.write( "That could happen at 2, 1, or zero points.<BR>\n" );
document.write( "At those points, if any, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2-3x%2B4=0 ALIGN=MIDDLE  ALT=\"y=3x%5E2-3x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The x-coordinates of those points are the real solutions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-3x%2B4=0 ALIGN=MIDDLE  ALT=\"3x%5E2-3x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since all real numbers are also complex numbers, those can be called complex solutions, but there is no imaginary part to them.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Every quadratic equation has \"two complex number solutions\",<BR>\n" );
document.write( "but there is a trick (coming) about that \"two\". <BR>\n" );
document.write( "If the graph touches the axis at just one point, as in the graph below, the solution is a double real solution. It counts as \"two\" real solutions, and there are no solutions with anything imaginary about them.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.1%2C0.9%2C-0.1%2C0.9%2C%28x-0.5%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-0.1%2C0.9%2C-0.1%2C0.9%2C%28x-0.5%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "If the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2-3x%2B4 ALIGN=MIDDLE  ALT=\"y=3x%5E2-3x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not touch or cross the x-axis at all,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-3x%2B4=0 ALIGN=MIDDLE  ALT=\"3x%5E2-3x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has two complex solutions of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+%2B-+i%2AI ALIGN=MIDDLE  ALT=\"R+%2B-+i%2AI\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with a non-zero imaginary part. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=I ALIGN=MIDDLE  ALT=\"I\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.1%2C0.9%2C-0.1%2C0.9%2C%0D%0Agraph%28300%2C300%2C-0.1%2C0.9%2C-0.1%2C0.9%2C%28x-0.5%29%5E2%2B0.3%29%2C%0D%0Aarrow%280.3%2C0.3%2C0.5%2C0.3%29%2Carrow%280.2%2C0.3%2C0%2C0.3%29%2C%0D%0Aarrow%280.5%2C0%2C0.5%2C0.3%29%2Carrow%280.5%2C0.3%2C0.5%2C0%29%2C%0D%0Alocate%280.23%2C0.32%2CR%29%2Clocate%280.51%2C0.2%2CI%5E2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "SOLVING WITH SOME CALCULATIONS:<BR>\n" );
document.write( "Quadratic equations can be written in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E3%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E3%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > are numbers called coefficients.<BR>\n" );
document.write( "The solutions are given by<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+=%28-b%29%2F%282%2Aa%29+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac%29%2F%282%2Aa%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With some calculation you know if all the solutions are real.<BR>\n" );
document.write( "You do not need to calculate the values of the solutions.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%3E4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2%3E4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the square root is a non-zero real number,<BR>\n" );
document.write( "and you have two different real solutions.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2=4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac=0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the square root is zero,<BR>\n" );
document.write( "and you have one (double) real solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%3C4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2%3C4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the square root is a purely imaginary number,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%2Asqrt%284%2Aa%2Ac-b%5E2%29 ALIGN=MIDDLE  ALT=\"i%2Asqrt%284%2Aa%2Ac-b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and you know that there are<BR>\n" );
document.write( "two conjugate complex solutions that can be written (both together) as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F%282%2Aa%29+%2B-+i%2AI ALIGN=MIDDLE  ALT=\"%28-b%29%2F%282%2Aa%29+%2B-+i%2AI\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "In your case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Aa%2Ac=4%2A3%2A4=48 ALIGN=MIDDLE  ALT=\"4%2Aa%2Ac=4%2A3%2A4=48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=%28-3%29%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=%28-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%3C4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2%3C4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you have no real solutions.<BR>\n" );
document.write( "We so not need to calculate <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aa=2%2A3=6 ALIGN=MIDDLE  ALT=\"2%2Aa=2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2Aa%2Ac-b%5E2%29=sqrt%2839%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2Aa%2Ac-b%5E2%29=sqrt%2839%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Aa%2Ac-b%5E2=48-9=39 ALIGN=MIDDLE  ALT=\"4%2Aa%2Ac-b%5E2=48-9=39\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2Aa%2Ac-b%5E2%29%2F%282%2Aa%29=sqrt%2839%29%2F6=I ALIGN=MIDDLE  ALT=\"sqrt%284%2Aa%2Ac-b%5E2%29%2F%282%2Aa%29=sqrt%2839%29%2F6=I\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or the solutions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+1%2F2+%2B-+i%2A%28sqrt%2839%29%2F6%29 ALIGN=MIDDLE  ALT=\"x=+1%2F2+%2B-+i%2A%28sqrt%2839%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F%282%2Aa%29=3%2F2%2A2=1%2F2=R ALIGN=MIDDLE  ALT=\"%28-b%29%2F%282%2Aa%29=3%2F2%2A2=1%2F2=R\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );