document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=195270>Question 195270</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis. <BR>\n" );
document.write( "y = 2x^2 + x + 3 <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 #146465 by </B><A HREF=/tutors/aboutme.mpl?userid=J2R2R><B>J2R2R(94)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=J2R2R><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=J2R2R><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=146465\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis. \r<BR>\n" );
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document.write( "y = 2x^2 + x + 3\r<BR>\n" );
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document.write( "Here, the discriminant will tell us how many real or imaginary roots we have.\r<BR>\n" );
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document.write( "b^2 – 4ac > 0 gives two real solutions which are different (crosses the x-axis in two places)\r<BR>\n" );
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document.write( "b^2 – 4ac = 0 gives two real solutions which are equal (crosses the x-axis in one place at the vertex)\r<BR>\n" );
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document.write( "b^2 – 4ac < 0 gives two complex solutions which are different (doesn’t cross the x-axis).\r<BR>\n" );
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document.write( "For our equation, a = 2, b = 1, c = 3\r<BR>\n" );
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document.write( "b^2 – 4ac = 1 – 24 = – 23\r<BR>\n" );
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document.write( "Thus we have two complex roots which are different and therefore the parabola doesn’t cross the x-axis so there are no points in common with the x-axis, and the vertex lies above the x-axis. </SPAN>\n" );
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