document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172054>Question 172054</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the additive inverse of each number: -4+6i.<BR>\n" );
document.write( "find the absolute value of each complex number: 6i<BR>\n" );
document.write( "solve each quadratic equation: -x^2+5x-2=0, x^2-8x+7=0\r<BR>\n" );
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document.write( "i know the first two are pretty simple, i'm just not sure exactly how to do them.<BR>\n" );
document.write( "and the last two i am completely confused on them :( </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127109 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=127109\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The additive inverse of any number is that number when added to the original number results in a sum of zero.  So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your original number, the additive inverse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a ALIGN=MIDDLE  ALT=\"-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B%28-a%29=0 ALIGN=MIDDLE  ALT=\"a%2B%28-a%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To sum two complex numbers, add the real parts and add the imaginary parts, thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%2B%28c%2Bdi%29=+%28a%2Bc%29%2B%28b%2Bd%29i ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%2B%28c%2Bdi%29=+%28a%2Bc%29%2B%28b%2Bd%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, in order to find the additive inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2B6i ALIGN=MIDDLE  ALT=\"-4%2B6i\"   DISABLED_style= \"max-width:90%; height:auto;\" > we need to determine the additive inverses of both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Hence the additive inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2B6i ALIGN=MIDDLE  ALT=\"-4%2B6i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-6i ALIGN=MIDDLE  ALT=\"4-6i\"   DISABLED_style= \"max-width:90%; height:auto;\" > because\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-4%2B6i%29%2B%284-6i%29=0%2B0i=0+ ALIGN=MIDDLE  ALT=\"%28-4%2B6i%29%2B%284-6i%29=0%2B0i=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The absolute value of a complex number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=%28x%2Byi%29 ALIGN=MIDDLE  ALT=\"z=%28x%2Byi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28z%29=sqrt%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28z%29=sqrt%28x%5E2%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In this case, you are looking for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%286i%29=abs%280%2B6i%29=sqrt%280%5E2%2B6%5E2%29=sqrt%2836%29=6 ALIGN=MIDDLE  ALT=\"abs%286i%29=abs%280%2B6i%29=sqrt%280%5E2%2B6%5E2%29=sqrt%2836%29=6\"   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=x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  are the two roots of a quadratic equation of the form <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;\" >\r<BR>\n" );
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document.write( "All you need to do is plug in the values of the coefficients in your quadratic equation and do the arithmetic.  Remember to use a factor of i if the part under the radical is less than zero.\r<BR>\n" );
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document.write( "For your first equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2 ALIGN=MIDDLE  ALT=\"c=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-5+%2B-+sqrt%28+5%5E2-4%28-1%29%28-2%29+%29%29%2F%282%28-1%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-5+%2B-+sqrt%28+5%5E2-4%28-1%29%28-2%29+%29%29%2F%282%28-1%29%29+\"   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=x+=+%28-5+%2B-+sqrt%28+25-8+%29%29%2F%28-2%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-5+%2B-+sqrt%28+25-8+%29%29%2F%28-2%29+\"   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=x+=+%28-5+%2B-+sqrt%28+17+%29%29%2F%28-2%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-5+%2B-+sqrt%28+17+%29%29%2F%28-2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=%285-sqrt%2817%29%29%2F2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=%285-sqrt%2817%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=%285%2Bsqrt%2817%29%29%2F2 ALIGN=MIDDLE  ALT=\"x%5B2%5D=%285%2Bsqrt%2817%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "These are the exact representations of the two roots of this quadratic.  If you want numerical approximations, use your calculator.  Seek guidance from your text or your instructor as to the required precision whenever numerical approximations are required.\r<BR>\n" );
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document.write( "The other problem is worked exactly the same way, and again has a positive discriminant, the second one also has real number roots.\r<BR>\n" );
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