document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1089048>Question 1089048</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for what value of a does the quadratic function f(x)=ax^2-6x+3 have no <BR>\n" );
document.write( "x-intercepts?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #703380 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=703380\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >-intercept is the point where a parabola crosses the x-axis. This point is also known as a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zero ALIGN=MIDDLE  ALT=\"zero\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root ALIGN=MIDDLE  ALT=\"root\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=solution ALIGN=MIDDLE  ALT=\"solution\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+%96+4ac ALIGN=MIDDLE  ALT=\"b%5E2+%96+4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >, called the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=discriminant ALIGN=MIDDLE  ALT=\"discriminant\"   DISABLED_style= \"max-width:90%; height:auto;\" >\", and we can use it to find out what kind and how many roots the quadratic function have \r<BR>\n" );
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document.write( "rule:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-4ac+%3E+0 ALIGN=MIDDLE  ALT=\"b%5E2+-4ac+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Discriminant is greater than zero, Positive Discriminant:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Two ALIGN=MIDDLE  ALT=\"Two\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Real ALIGN=MIDDLE  ALT=\"+Real\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solutions\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Discriminant is equal to zero. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=One ALIGN=MIDDLE  ALT=\"One\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Real ALIGN=MIDDLE  ALT=\"+Real\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solution\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-4ac+%3C+0 ALIGN=MIDDLE  ALT=\"b%5E2+-4ac+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Discriminant is less than zero ,Negative Discriminant: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=No ALIGN=MIDDLE  ALT=\"No\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Real ALIGN=MIDDLE  ALT=\"+Real\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solutions => means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >-intercepts\r<BR>\n" );
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document.write( "so, you will use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-4ac+%3C+0 ALIGN=MIDDLE  ALT=\"b%5E2+-4ac+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E2-6x%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E2-6x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=a ALIGN=MIDDLE  ALT=\"a=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%5E2+-4a%2A3%3C+0 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2+-4a%2A3%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=36+-12a%3C+0 ALIGN=MIDDLE  ALT=\"36+-12a%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=36+%3C+12a ALIGN=MIDDLE  ALT=\"36+%3C+12a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2F12+%3C+a ALIGN=MIDDLE  ALT=\"36%2F12+%3C+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E3 ALIGN=MIDDLE  ALT=\"a%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "check:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E3 ALIGN=MIDDLE  ALT=\"a%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can use <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;\" > and see the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=4x%5E2-6x%2B3 ALIGN=MIDDLE  ALT=\"f%28x%29=4x%5E2-6x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+4x%5E2-6x%2B3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+4x%5E2-6x%2B3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "as you can see, there is no x-intercept\r<BR>\n" );
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