document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=962773>Question 962773</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi tutor ! So im a 14 year old who lives in texas . Ive really been struggling with the quadratic formula and would love for you to help me where i need help . Please please please respond here , on my email , or through text 2144317404 ! Thank you and God bless you . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #588264 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=588264\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The quadratic formula is about a \"general\"<BR>\n" );
document.write( "quadratic equation, namely:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then you set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29+=+0+ ALIGN=MIDDLE  ALT=\"+f%28x%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the \"roots\"'<BR>\n" );
document.write( "which are the solutions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+ ALIGN=MIDDLE  ALT=\"+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "Suppose your equation for finding roots looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-4x%5E2+%2B+6x+-+11+=+0+ ALIGN=MIDDLE  ALT=\"+-4x%5E2+%2B+6x+-+11+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, comparing to the \"general\" equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+-4+ ALIGN=MIDDLE  ALT=\"+a+=+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+-11+ ALIGN=MIDDLE  ALT=\"+c+=+-11+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "Now you have to plug these values into the \"general\"<BR>\n" );
document.write( "formula for finding the roots:<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( "The tricky thing about this formula is that it is TWO formulas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-b+%2B+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%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( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-b+-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-b+-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-------------------------------------<BR>\n" );
document.write( "That's because of the + or - in front of the square root sign<BR>\n" );
document.write( "---------------------------<BR>\n" );
document.write( "Now plug in values:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+6%5E2+-+4%2A%28-4%29%2A%28-11%29+%29%29%2F%282%2A%28-4%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+6%5E2+-+4%2A%28-4%29%2A%28-11%29+%29%29%2F%282%2A%28-4%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+36+-+176+%29%29%2F%28-8%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+36+-+176+%29%29%2F%28-8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28+-6+%2B-sqrt%28+-140+%29%29+%2F+%28-8%29+ ALIGN=MIDDLE  ALT=\"+x+=+%28+-6+%2B-sqrt%28+-140+%29%29+%2F+%28-8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B1%5D+=+%28+-6+%2B+11.8322i+%29+%2F+%28-8%29+ ALIGN=MIDDLE  ALT=\"+x%5B1%5D+=+%28+-6+%2B+11.8322i+%29+%2F+%28-8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B1%5D+=+3%2F4+-+1.4790i+ ALIGN=MIDDLE  ALT=\"+x%5B1%5D+=+3%2F4+-+1.4790i+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is one complex solution<BR>\n" );
document.write( "The other one is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B2%5D+=+%28+-6+-+11.8322i+%29+%2F+%28-8%29+ ALIGN=MIDDLE  ALT=\"+x%5B2%5D+=+%28+-6+-+11.8322i+%29+%2F+%28-8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B2%5D+=+3%2F4+%2B+1.4790i+ ALIGN=MIDDLE  ALT=\"+x%5B2%5D+=+3%2F4+%2B+1.4790i+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( "So, you ended up with what they call a pair of<BR>\n" );
document.write( "complex conjugate roots. That just means they<BR>\n" );
document.write( "look like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+%2B+Bi+ ALIGN=MIDDLE  ALT=\"+A+%2B+Bi+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+-+Bi+ ALIGN=MIDDLE  ALT=\"+A+-+Bi+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------------------<BR>\n" );
document.write( "Then there is the whole discussion of the \"discriminant\"<BR>\n" );
document.write( "which is the <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;\" > part of the formula<BR>\n" );
document.write( "Because this is under the square root sign, the<BR>\n" );
document.write( "\"discriminant\" will tell you whether you have:<BR>\n" );
document.write( "(1) one \"double\" root in the case where:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+=+4a%2Ac+ ALIGN=MIDDLE  ALT=\"+b%5E2+=+4a%2Ac+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) two rational roots when:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+%3E+4a%2Ac+ ALIGN=MIDDLE  ALT=\"+b%5E2+%3E+4a%2Ac+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) that pair of complex conjugate roots where:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+%3C+4a%2Ac+ ALIGN=MIDDLE  ALT=\"+b%5E2+%3C+4a%2Ac+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------------------<BR>\n" );
document.write( "The picture of the quadratic ( parabola ) tells you right<BR>\n" );
document.write( "away which case you have:<BR>\n" );
document.write( "(1) the parabola just touches the x-axis<BR>\n" );
document.write( "(2) the parabola crosses the x-axis in 2 places<BR>\n" );
document.write( "(3) the parabola never touches the x-axis<BR>\n" );
document.write( "---------------------------------------<BR>\n" );
document.write( "Here's a plot of the 3 cases of the discriminant:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-5%2C+5%2C+-5%2C+5%2C+2x%5E2+%2B+4x+%2B+2%2C+2x%5E2+%2B+3x+%2B+2%2C+2x%5E2+%2B+5x+%2B+2++%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Hope this helps\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );