document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=541963>Question 541963</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The question from my book says that \"the radicand of the quadratic formula, b^2 - 4ac, can be used to determine whether <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ax%5E2+%2B+bx+%2B+c+=+a ALIGN=MIDDLE  ALT=\"+ax%5E2+%2B+bx+%2B+c+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > has solutions that are rational, irrational, or not real numbers.\" I haven't been able to solve the problem on this website because it's saying that the first coefficient is zero. It's also asking me to explain how this works. And if it possible to determine the kinds of answers that one will obtain to a quadratic equation without actually solving the equation. I would greatly appreciate any help I can get. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354376 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=354376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The quadratic formula is:<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( "I will make the replacement<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28b%5E2++-+4a%2Ac%29+=+z+ ALIGN=MIDDLE  ALT=\"+sqrt%28b%5E2++-+4a%2Ac%29+=+z+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28-b+%2B+z%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x+=+%28-b+%2B+z%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+=+%28-b+-+z%29+%2F+%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"+x+=+%28-b+-+z%29+%2F+%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not zero and is a real number, I<BR>\n" );
document.write( "get a different solution for each equation,<BR>\n" );
document.write( "so 2 solutions.<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+0 ALIGN=MIDDLE  ALT=\"z+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then there is only 1 solution<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "Also, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a square root, it can be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28-1%2An%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28-1%2An%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+i%2Asqrt%28n%29+ ALIGN=MIDDLE  ALT=\"+i%2Asqrt%28n%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and there are 2 imaginary solutions.<BR>\n" );
document.write( "-------------------------------<BR>\n" );
document.write( "These possibilities depend on what<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;\" > is<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;\" > = 0 ( one solution )<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;\" > = (+) ( two real solutions )<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;\" > = (-) ( two imaginary solutions )<BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "In the case of  <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;\" > = (+), if this is a perfect square,<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+64+ ALIGN=MIDDLE  ALT=\"+64+\"   DISABLED_style= \"max-width:90%; height:auto;\" > The 2 real solutions will be rational, which means<BR>\n" );
document.write( "a fraction of some kind.<BR>\n" );
document.write( "If it is not a perfect square, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31 ALIGN=MIDDLE  ALT=\"31\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the 2 real solutions<BR>\n" );
document.write( "will be irrational, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3.666666+ ALIGN=MIDDLE  ALT=\"+3.666666+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. . . forever<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "Hope this isn't confusing. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );