document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=587200>Question 587200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good day to you all. \r<BR>\n" );
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document.write( "Please determine the nature of the solutions to each equation without solving it. \r<BR>\n" );
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document.write( "1.) 25x^2 - 10x + 1 = 0<BR>\n" );
document.write( "2.) x^2 + x +2 = 0<BR>\n" );
document.write( "3.) 5x^2 - x = 0\r<BR>\n" );
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document.write( "Thank You.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #374023 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=374023\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Look at <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;\" > where the<BR>\n" );
document.write( "equation has the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2Ax%5E2+%2B+b%2Ax+%2Bc+=+0+ ALIGN=MIDDLE  ALT=\"+a%2Ax%5E2+%2B+b%2Ax+%2Bc+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+-+4%2Aa%2Ac+=+%28-10%29%5E2+-+4%2A25%2A1+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+%28-10%29%5E2+-+4%2A25%2A1+\"   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+-+4%2Aa%2Ac+=+100+-+100+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+100+-+100+\"   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+-+4%2Aa%2Ac+=+0+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This means there is 1 real solution<BR>\n" );
document.write( "It is called a double root<BR>\n" );
document.write( "---------------------------------<BR>\n" );
document.write( "(2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+-+4%2Aa%2Ac+=+1%5E2+-+4%2A1%2A2+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+1%5E2+-+4%2A1%2A2+\"   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+-+4%2Aa%2Ac+=+1+-+8+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+1+-+8+\"   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+-+4%2Aa%2Ac+=+-7+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+-7+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is minus, so there are 2<BR>\n" );
document.write( "imaginary roots ( look like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Ai ALIGN=MIDDLE  ALT=\"a%2Bb%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-b%2Ai ALIGN=MIDDLE  ALT=\"a-b%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "-----------------------------------<BR>\n" );
document.write( "(3)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+-+4%2Aa%2Ac+=+%28-1%29%5E2+-+4%2A5%2A0+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+%28-1%29%5E2+-+4%2A5%2A0+\"   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+-+4%2Aa%2Ac+=+1+ ALIGN=MIDDLE  ALT=\"+b%5E2+-+4%2Aa%2Ac+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is plus, so there are 2 real roots<BR>\n" );
document.write( "-------------------------------------<BR>\n" );
document.write( "Here's plots of the 3 equations.<BR>\n" );
document.write( "Touching the x-axis: 1 solution<BR>\n" );
document.write( "Not touching at all: 2 imaginary solutions<BR>\n" );
document.write( "Touching twice: 2 real solutions<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-2%2C+2%2C+-2%2C+5%2C+25x%5E2-10x%2B1%2Cx%5E2%2Bx%2B2%2C5x%5E2-x+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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