document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=48581>Question 48581</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use the discriminant to determine the number of solutions of the quadratic equation, and whether the solutions are real or complex.  Note: It is not necessary to find the roots; just determine the number and types of solutions.\r<BR>\n" );
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document.write( " 2x^2 + x - 1 = 0\r<BR>\n" );
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document.write( " 4/3x^2 - 2x + 3/4 = 0\r<BR>\n" );
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document.write( " 2x^2 + 5x + 5 = 0\r<BR>\n" );
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document.write( " 3z^2 + z - 1 = 0\r<BR>\n" );
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document.write( " m^2 + m + 1 = 0<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 #32166 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1806)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=32166\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 5 color = \"blue\"><b>use the discriminant to determine the number of solutions\r\n" );
document.write( "of the quadratic equation, and whether the solutions are \r\n" );
document.write( "real or complex. Note: It is not necessary to find the \r\n" );
document.write( "roots; just determine the number and types of solutions. \r\n" );
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document.write( "Just learn these rules:\r\n" );
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document.write( "Ax² + Bx + C =0 (or any letter besides x)\r\n" );
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document.write( "has the discriminant  B² - 4AC. Calculate it.\r\n" );
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document.write( "1. If the discriminant is 0, there is one real solution.\r\n" );
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document.write( "2. If the discriminant is negative, there are two conjugate\r\n" );
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document.write( "3. If the discriminant is positive, there are two real \r\n" );
document.write( "   solutions. \r\n" );
document.write( "   A. If the discriminant is a perfect square, the two real \r\n" );
document.write( "      solutions are rational.\r\n" );
document.write( "   B. If the discriminant is not a perfect square, the two \r\n" );
document.write( "      real solutions are irrational.\r\n" );
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document.write( "2x² + x - 1 = 0\r\n" );
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document.write( "Give the x a coefficient of 1:\r\n" );
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document.write( "2x² + 1x - 1 = 0\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "Ax² + Bx + C = 0\r\n" );
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document.write( "A = 2, B = 1, and C = -1\r\n" );
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document.write( "Calculate discriminant     B² - 4AC\r\n" );
document.write( "                       (1)² - 4(2)(-1)\r\n" );
document.write( "                          1 + 8\r\n" );
document.write( "                            9\r\n" );
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document.write( "9 is positive, so there by rule 3 above there are two real\r\n" );
document.write( "solutions.  9 is also a perfect square, so by rule 3A the \r\n" );
document.write( "two real solutions are rational. \r\n" );
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document.write( "4/3x² - 2x + 3/4 = 0 \r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "Ax² + Bx + C = 0\r\n" );
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document.write( "A = 4/3, B = -2, and C = 3/4\r\n" );
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document.write( "Calculate discriminant     B² - 4AC\r\n" );
document.write( "                       (-2)² - 4(4/3)(3/4)\r\n" );
document.write( "                           -4\r\n" );
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document.write( "-4 is negative, so there by rule 2 above there are \r\n" );
document.write( "two conjugate complex solutions.\r\n" );
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document.write( "2x² + 5x + 5 = 0 \r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "Ax² + Bx + C = 0\r\n" );
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document.write( "A = 2, B = 5, and C = 5\r\n" );
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document.write( "Calculate discriminant     B² - 4AC\r\n" );
document.write( "                        (5)² - 4(2)(5)\r\n" );
document.write( "                          25 - 40\r\n" );
document.write( "                            -13\r\n" );
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document.write( "-13 is negative, so there by rule 2 above there are \r\n" );
document.write( "two conjugate complex solutions.\r\n" );
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document.write( "3z² + z - 1 = 0\r\n" );
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document.write( "Give the z a coefficient of 1\r\n" );
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document.write( "3z² + 1z - 1 = 0\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "Ax² + Bx + C = 0\r\n" );
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document.write( "A = 3, B = 1, and C = -1\r\n" );
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document.write( "Calculate discriminant     B² - 4AC\r\n" );
document.write( "                       (1)² - 4(3)(-1)\r\n" );
document.write( "                          1 + 12\r\n" );
document.write( "                            13\r\n" );
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document.write( "13 is positive, so there by rule 3 above there are two real\r\n" );
document.write( "solutions.  13 is not a perfect square, so by rule 3B the \r\n" );
document.write( "two real solutions are irrational. \r\n" );
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document.write( "m² + m + 1 = 0\r\n" );
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document.write( "Give the m² and the m coeficients of 1 each\r\n" );
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document.write( "1m² + 1m + 1 = 0\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "Ax² + Bx + C = 0\r\n" );
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document.write( "A = 1, B = 1, and C = 1\r\n" );
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document.write( "Calculate discriminant     B² - 4AC\r\n" );
document.write( "                        (1)² - 4(1)(1)\r\n" );
document.write( "                           1 - 4\r\n" );
document.write( "                            -3\r\n" );
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document.write( "-3 is negative, so there by rule 2 above there are \r\n" );
document.write( "two conjugate complex solutions.\r\n" );
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document.write( "Edwin</PRE>\r<BR>\n" );
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