document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207475>Question 1207475</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4 - i is a solution of a quadratic equation with real coefficients.Find the other solution.\r<BR>\n" );
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document.write( "How is this one done?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845370 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=845370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>4+i</font>\r<BR>\n" );
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document.write( "Quick explanation:  If a+bi is one root, then its paired counterpart (known as the complex conjugate) would be a-bi<BR>\n" );
document.write( "This applies only when all of the coefficients are real numbers.\r<BR>\n" );
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document.write( "Another example: The root 2+i pairs up with 2-i\r<BR>\n" );
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document.write( "Let's say you didn't know about the complex conjugate, or that you might be curious about an alternative pathway.\r<BR>\n" );
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document.write( "We can isolate the \"i\" term and square both sides to generate a quadratic from it.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4+=+-i ALIGN=MIDDLE  ALT=\"x-4+=+-i\"   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=%28x-4%29%5E2+=+%28-i%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+%28-i%29%5E2\"   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=%28x-4%29%5E2+=+-1 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+=+-1\"   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%5E2-8x%2B16+=+-1 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16+=+-1\"   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%5E2-8x%2B16%2B1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2B1+=+0\"   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%5E2-8x%2B17+=+0 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B17+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's apply the quadratic formula<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%281%29%2817%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%281%29%2817%29%29%29%2F%282%281%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+=+%288%2B-sqrt%2864+-+68%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%288%2B-sqrt%2864+-+68%29%29%2F%282%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+=+%288%2B-sqrt%28-4%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%288%2B-sqrt%28-4%29%29%2F%282%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+=+%288%2B-+2i%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%288%2B-+2i%29%2F%282%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+=+%288%2B+2i%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%288%2B+2i%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%288-2i%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%288-2i%29%2F%282%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+=+4%2Bi ALIGN=MIDDLE  ALT=\"x+=+4%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4-i ALIGN=MIDDLE  ALT=\"x+=+4-i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We arrive at roots <font color=red>4+i</font> and 4-i to help confirm the answer. \r<BR>\n" );
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document.write( "This is a somewhat long-winded pathway to basically rephrase what I mentioned at the top. <BR>\n" );
document.write( "If a+bi is one root, then a-bi is also included in the mix. <BR>\n" );
document.write( "This only applies when all coefficients are real numbers. <BR>\n" );
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