document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1075729>Question 1075729</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what real values of a does the quadratic 4x^2 + ax + 25 have non-real roots? Give your answer as an interval.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #690368 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=690368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The quadratic formula says that the roots are:<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( "when the equation has the form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+ ALIGN=MIDDLE  ALT=\"+y+=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "They decided that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+a+ ALIGN=MIDDLE  ALT=\"+b+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( a little confusing )<BR>\n" );
document.write( "so the list is:<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+=+a+ ALIGN=MIDDLE  ALT=\"+b+=+a+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+25+ ALIGN=MIDDLE  ALT=\"+c+=+25+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I can say:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28+-a+%2B-+sqrt%28+a%5E2-4%2A4%2A25+%29%29%2F%282%2A4%29+ ALIGN=MIDDLE  ALT=\"+x+=+%28+-a+%2B-+sqrt%28+a%5E2-4%2A4%2A25+%29%29%2F%282%2A4%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+-a+%2B-+sqrt%28+a%5E2+-+400+%29%29%2F8+ ALIGN=MIDDLE  ALT=\"+x+=+%28+-a+%2B-+sqrt%28+a%5E2+-+400+%29%29%2F8+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The roots will be non-real ( imaginary ) whenever<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2+-+400+ ALIGN=MIDDLE  ALT=\"+a%5E2+-+400+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is negative. ( this is called the discriminant )<BR>\n" );
document.write( "So I can say:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2+-+400+%3C+0+ ALIGN=MIDDLE  ALT=\"+a%5E2+-+400+%3C+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2+%3C+400+ ALIGN=MIDDLE  ALT=\"+a%5E2+%3C+400+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+%3C+20+ ALIGN=MIDDLE  ALT=\"+a+%3C+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and also:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+%3E+-20+ ALIGN=MIDDLE  ALT=\"+a+%3E+-20+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Putting these together:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-20+%3C+a+%3C+20++ ALIGN=MIDDLE  ALT=\"+-20+%3C+a+%3C+20++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+0+ ALIGN=MIDDLE  ALT=\"+a+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfies the conditions, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28+-a+%2B-+sqrt%28+a%5E2+-+400+%29%29%2F8+ ALIGN=MIDDLE  ALT=\"+x+=+%28+-a+%2B-+sqrt%28+a%5E2+-+400+%29%29%2F8+\"   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+-0+%2B-+sqrt%28+0%5E2+-+400+%29%29%2F8+ ALIGN=MIDDLE  ALT=\"+x+=+%28+-0+%2B-+sqrt%28+0%5E2+-+400+%29%29%2F8+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%281%2F8%29%2A%28+sqrt%28+-400+%29+%29+ ALIGN=MIDDLE  ALT=\"+x+=+%281%2F8%29%2A%28+sqrt%28+-400+%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+=+%281%2F8%29%2A%28+20i+%29+ ALIGN=MIDDLE  ALT=\"+x+=+%281%2F8%29%2A%28+20i+%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+=+%285%2F2%29%2Ai+ ALIGN=MIDDLE  ALT=\"+x+=+%285%2F2%29%2Ai+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and also:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%281%2F8%29%2A%28+-sqrt%28+-400+%29+%29+ ALIGN=MIDDLE  ALT=\"+x+=+%281%2F8%29%2A%28+-sqrt%28+-400+%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+=+%28-5%2F2%29%2Ai+ ALIGN=MIDDLE  ALT=\"+x+=+%28-5%2F2%29%2Ai+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Here is the plot when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+0+ ALIGN=MIDDLE  ALT=\"+a+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+400%2C+400%2C+-6%2C+6%2C+-5%2C+50%2C+4x%5E2+%2B+25+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+400%2C+400%2C+-6%2C+6%2C+-5%2C+50%2C+4x%5E2+%2B+25+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );