document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=102279>Question 102279</A>: <I><SPAN STYLE=\"word-break: break-all;\"> decide all values of b in the following equations that will<BR>\n" );
document.write( "give one or more real number solutions.<BR>\n" );
document.write( "(a) 3x^2 + bx - 3 =
 0<BR>\n" );
document.write( "(b) 5x^2 + bx + 1 =
 0<BR>\n" );
document.write( "(c) -3x^2 + bx - 3 =
 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #74331 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=74331\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In the standard form <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2Bbx+%2Bc+=+0++ ALIGN=MIDDLE  ALT=\"ax%5E2+%2Bbx+%2Bc+=+0++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the roots are:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x=-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if 	<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2-4ac%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28b%5E2-4ac%29%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " then the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=real ALIGN=MIDDLE  ALT=\"real\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Otherwise, the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=imaginary ALIGN=MIDDLE  ALT=\"imaginary\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
document.write( "\n" );
document.write( "So<BR>\n" );
document.write( "(a)	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2+-+4%2A3%2A3%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28b%5E2+-+4%2A3%2A3%29%3E=+0\"   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+%3E=36 ALIGN=MIDDLE  ALT=\"+b%5E2+%3E=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%3E=+6+ ALIGN=MIDDLE  ALT=\"+b%3E=+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=real ALIGN=MIDDLE  ALT=\"real\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any vale of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+6 ALIGN=MIDDLE  ALT=\"b+%3E=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "(b)	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2+-+4%2A5%2A1%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28b%5E2+-+4%2A5%2A1%29%3E=+0\"   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+%3E=%284%2A5%29 ALIGN=MIDDLE  ALT=\"b%5E2+%3E=%284%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+sqrt%284%2A5%29 ALIGN=MIDDLE  ALT=\"b+%3E=+sqrt%284%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+2%2A+sqrt%285%29 ALIGN=MIDDLE  ALT=\"b+%3E=+2%2A+sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=real ALIGN=MIDDLE  ALT=\"real\"   DISABLED_style= \"max-width:90%; height:auto;\" >for any vale of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+2%2A+sqrt+%285%29 ALIGN=MIDDLE  ALT=\"b+%3E=+2%2A+sqrt+%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "(c)	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%5E2+-+%284%2A-3%2A-3%29%29%3E=+0 ALIGN=MIDDLE  ALT=\"%28b%5E2+-+%284%2A-3%2A-3%29%29%3E=+0\"   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+%3E=+36 ALIGN=MIDDLE  ALT=\"b%5E2+%3E=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+sqrt+%2836%29 ALIGN=MIDDLE  ALT=\"b+%3E=+sqrt+%2836%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=++6 ALIGN=MIDDLE  ALT=\"b+%3E=++6\"   DISABLED_style= \"max-width:90%; height:auto;\" >    the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=real ALIGN=MIDDLE  ALT=\"real\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+%3E=+6 ALIGN=MIDDLE  ALT=\"b+%3E=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );