document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=253132>Question 253132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the quadratic equation whose roots have a sum of 12 and the roots difference is 4i. (Write a let statement to identify your roots) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #185395 by </B><A HREF=/tutors/aboutme.mpl?userid=MathTherapy><B>MathTherapy(10552)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathTherapy><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=MathTherapy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=185395\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find any quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have to realize that:\r<BR>\n" );
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document.write( "a = 1 (always)<BR>\n" );
document.write( "b = - (sum of roots)<BR>\n" );
document.write( "c = product of roots\r<BR>\n" );
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document.write( "Since a is always 1, and the sum of the roots = 12, then b = - (12) = -12\r<BR>\n" );
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document.write( "We now have: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+12x+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+12x+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "To find c, we need to 1st determine the roots and multiply them\r<BR>\n" );
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document.write( "Let root 1 be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and root 2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since the sum of the roots = 12, then ----- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D+%2B+r%5B2%5D+=+12 ALIGN=MIDDLE  ALT=\"r%5B1%5D+%2B+r%5B2%5D+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --------- eq (i)<BR>\n" );
document.write( "Also, since the roots’ difference = 4i, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D+-+r%5B2%5D+=+4i ALIGN=MIDDLE  ALT=\"r%5B1%5D+-+r%5B2%5D+=+4i\"   DISABLED_style= \"max-width:90%; height:auto;\" > --------- eq (ii)\r<BR>\n" );
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document.write( "Adding equations (i) & (ii), we get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r%5B1%5D+=+12+%2B+4i ALIGN=MIDDLE  ALT=\"2r%5B1%5D+=+12+%2B+4i\"   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=r%5B1%5D+=+%2812+%2B+4i%29%2F2 ALIGN=MIDDLE  ALT=\"r%5B1%5D+=+%2812+%2B+4i%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D+=+6+%2B+2i ALIGN=MIDDLE  ALT=\"r%5B1%5D+=+6+%2B+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Substituting 6 + 2i for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > in eq (i), we get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+%2B+2i+%2B+r%5B2%5D+=+12 ALIGN=MIDDLE  ALT=\"6+%2B+2i+%2B+r%5B2%5D+=+12\"   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=r%5B2%5D+=+12+-+6+-+2i ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+12+-+6+-+2i\"   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=r%5B2%5D+=+6+-+2i ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+6+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Since we now have both roots, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D+=+6+%2B+2i ALIGN=MIDDLE  ALT=\"r%5B1%5D+=+6+%2B+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D+=+6+-+2i ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+6+-+2i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we multiply these two roots to get “c.”\r<BR>\n" );
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document.write( "Therefore, “c” = (6 + 2i)(6 – 2i) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36+-+4i%5E2 ALIGN=MIDDLE  ALT=\"36+-+4i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or, 36 – 4(-1) = 40\r<BR>\n" );
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document.write( "With “a” being 1, “b” being – 12, and “c” being 40, the quadratic equation in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28x%5E2+-+12x+%2B+40+=+0%29 ALIGN=MIDDLE  ALT=\"highlight_green%28x%5E2+-+12x+%2B+40+=+0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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