document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=51310>Question 51310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> When using the quadratic formula to solve a quadratic equation (ax2 + bx + c = 0), the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative.\r<BR>\n" );
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document.write( "Create three unique equations where the discriminant is positive, zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx + c. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #34248 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=34248\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Notice the THREE parts to this solution:\r<BR>\n" );
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document.write( "CASE I:<BR>\n" );
document.write( "Discriminant is positive:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4+=0 ALIGN=MIDDLE  ALT=\"x%5E2-4+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <BR>\n" );
document.write( "There are TWO real solutions.  The implication for the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2+-4 ALIGN=MIDDLE  ALT=\"y=x%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is that the graph crosses the x axis at TWO points.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2-4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "CASE II:<BR>\n" );
document.write( "Discriminant is zero:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x%2B4+=0 ALIGN=MIDDLE  ALT=\"x%5E2-4x%2B4+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <BR>\n" );
document.write( "There is ONLY ONE real solution.  The implication for the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2+-4x%2B4 ALIGN=MIDDLE  ALT=\"y=x%5E2+-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is that the graph touches the x axis at only one point, but it does NOT cross the x axis.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2-4x%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2-4x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "CASE III:<BR>\n" );
document.write( "Discriminant is negative:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4+=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <BR>\n" );
document.write( "There are NO real solutions.  The implication for the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2+%2B4 ALIGN=MIDDLE  ALT=\"y=x%5E2+%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is that the graph never touches nor crosses the x axis.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-6%2C6%2C-6%2C6%2Cx%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is a VERY important concept, especially with graphing calculators!!  Congratulations on an excellent question!!\r<BR>\n" );
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document.write( "R^2 at SCC<BR>\n" );
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