document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=71936>Question 71936</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. <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 #51437 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=51437\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the discriminant is greater than zero, then you will have a positive square root. This means you will have 2 roots that are real numbers. <BR>\n" );
document.write( "Ex:  x^2+5x+6 if you put this into the quadratic equation you get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5%2B-sqrt%2825-24%29%29%2F2=%28-5%2B-sqrt%281%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28-5%2B-sqrt%2825-24%29%29%2F2=%28-5%2B-sqrt%281%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > This further shows that the roots are real numbers of x=-2 and x=-3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B5x%2B6+%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B5x%2B6+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >2 real roots\r<BR>\n" );
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document.write( "If the discriminant is equal to zero, then you will have only 1 root with multiplicity of 2. <BR>\n" );
document.write( "Ex: x^2+4x+4 place into quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-4%2B-sqrt%2816-16%29%29%2F2=%28-5%2B-sqrt%280%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28-4%2B-sqrt%2816-16%29%29%2F2=%28-5%2B-sqrt%280%29%29%2F2\"   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+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B4x%2B4+%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2B4x%2B4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 1 real root with multiplicity of 2\r<BR>\n" );
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document.write( "If you have a negative discriminant, then no real roots will result since it's not  possible to take the square root of a negative number and get a real result.<BR>\n" );
document.write( "Ex: x^2+x+3 if placed into the quadratic formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5%2B-sqrt%281-12%29%29%2F2=%28-5%2B-sqrt%28-11%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28-5%2B-sqrt%281-12%29%29%2F2=%28-5%2B-sqrt%28-11%29%29%2F2\"   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+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2Bx%2B3+%29 ALIGN=MIDDLE  ALT=\"graph%28+300%2C+200%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E2%2Bx%2B3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >No real roots, only complex roots<BR>\n" );
document.write( "So a you will have a graph that doesn't cross the x-axis but will have complex roots </SPAN>\n" );
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