document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84020>Question 84020</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" );
document.write( "\n" );
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" );
document.write( "\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 #60511 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=60511\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is an EXCELLENT question!  When you solve the quadratic equation <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;\" > if the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is POSITIVE, then there will be TWO distinct real roots.  If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NEGATIVE, then there will be NO real roots. If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is ZERO, then there will be only ONE real root, of multiplicity 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now, if you graph the quadratic function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2+%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2+%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the ROOTS of the quadratic equation described in the previous paragraph correspond to the ZEROS (that is, the x-intercepts) of the graph of the quadratic function.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Likewise, the NUMBER of roots in the quadratic equation corresponds to the NUMBER of x-intercepts of the quadratic function.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the quadratic function is POSITIVE, then there will be TWO x-intercepts.  For example: <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;\" >, the discriminant is POSITIVE, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+0%5E2+-4%2A1%2A%28-4%29=16 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+0%5E2+-4%2A1%2A%28-4%29=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  There are TWO x-intercepts at x= 2 and at x= -2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2-4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the quadratic function is NEGATIVE, then there will be NO x-intercepts.  For example: <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;\" >, the discriminant is NEGATIVE, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+0%5E2+-4%2A1%2A4=-16 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+0%5E2+-4%2A1%2A4=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Notice that there are NO x-intercepts, since the graph does not touch or cross the x-axis.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2%2B4%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is ZERO, then there will be only ONE x-intercept, and it will be of multiplicity 2.  For example: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the discriminant is ZERO <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac+=+0%5E2+-4%2A1%2A0=0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac+=+0%5E2+-4%2A1%2A0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  There is ONE x-intercept  at x= 0, and it has a multiplicity of 2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "R^2 at SCC\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );