document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187572>Question 187572</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the i for in some of the quadratic equations? like one of the answers to a qustion is x=3+-2i. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140591 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=140591\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ sqrt{-1}\">\r<BR>\n" );
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document.write( "is not defined in the real numbers.  In order to deal with a radicand that is less than zero, a non-Real number, <i>i</i> (for imaginary), was defined as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i^2 = -1\">.  This number, along with any necessary real coefficients, comprise the Complex numbers.  Complex numbers are of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a + bi\"> where <i>i</i> is the imaginary number and <i>a</i> and <i>b</i> are Real numbers.\r<BR>\n" );
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document.write( "Any time you use the quadratic formula and the discriminant (the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^2-4ac\"> under the radical) is less than zero, you need to factor out a -1, the square root of which is <i>i</i>, and then you can deal with the real number root of whatever is left under the radical.\r<BR>\n" );
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document.write( "The answer you gave, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3 \pm 2i\"> are the roots of:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 6x + 13 = 0\">\r<BR>\n" );
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document.write( "I know this because I multiplied:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - (3 + 2i))(x - (3 - 2i)) = x^2 - (3 + 2i)x - (3 - 2i)x + (3 + 2i)(3 - 2i) = x^2 - 6x + 9 + 4\">\r<BR>\n" );
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document.write( "So, using the quadratic formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{-(-6) \pm sqrt{36-52}}{2}=\frac{6 \pm sqrt{-16}}{2} = \frac{6 \pm sqrt{(-1)(4)(4)}}{2} = \frac {6 \pm 4i}{2} = 3 \pm 2i\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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