document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172153>Question 1172153</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the quadratic equation in x with the given roots 1±3i√5 / 2 <- (it is a fraction). Show clear solution thanksss! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #797076 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=797076\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Use the zero product property to help go from the given roots to a quadratic polynomial.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2B3i%2Asqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%281%2B3i%2Asqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281-3i%2Asqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%281-3i%2Asqrt%285%29%29%2F2\"   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=2x+=+1%2B3i%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2x+=+1%2B3i%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+=+1-3i%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2x+=+1-3i%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-1+=+3i%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2x-1+=+3i%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-1+=+-3i%2Asqrt%285%29 ALIGN=MIDDLE  ALT=\"2x-1+=+-3i%2Asqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%5E2+=+%283i%2Asqrt%285%29%29%5E2 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2+=+%283i%2Asqrt%285%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%5E2+=+%28-3i%2Asqrt%285%29%29%5E2 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2+=+%28-3i%2Asqrt%285%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%5E2+=+9i%5E2%2A5 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2+=+9i%5E2%2A5\"   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=%282x-1%29%5E2+=+9%28-1%29%2A5 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2+=+9%28-1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use i^2 = -1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%5E2+=+-45 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2+=+-45\"   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=%282x-1%29%5E2%2B45+=+0 ALIGN=MIDDLE  ALT=\"%282x-1%29%5E2%2B45+=+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=4x%5E2-4x%2B1%2B45+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2-4x%2B1%2B45+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-4x%2B46+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2-4x%2B46+=+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=2%282x%5E2-2x%2B23%29+=+0 ALIGN=MIDDLE  ALT=\"2%282x%5E2-2x%2B23%29+=+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=2x%5E2-2x%2B23+=+0%2F2 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23+=+0%2F2\"   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=2x%5E2-2x%2B23+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > This is the final answer.\r<BR>\n" );
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document.write( "We can use the quadratic formula to confirm this\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have a = 2, b = -2, c = 23\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%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+=+%282%2Bsqrt%28-180%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282%2Bsqrt%28-180%29%29%2F%282%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282-sqrt%28-180%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282-sqrt%28-180%29%29%2F%282%282%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+=+%282%2Bsqrt%2836%2A%28-1%29%2A5%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282%2Bsqrt%2836%2A%28-1%29%2A5%29%29%2F%282%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282-sqrt%2836%2A%28-1%29%2A5%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282-sqrt%2836%2A%28-1%29%2A5%29%29%2F%282%282%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+=+%282%2Bsqrt%2836%29%2Asqrt%28-1%29%2Asqrt%285%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282%2Bsqrt%2836%29%2Asqrt%28-1%29%2Asqrt%285%29%29%2F%282%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282-sqrt%2836%29%2Asqrt%28-1%29%2Asqrt%285%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282-sqrt%2836%29%2Asqrt%28-1%29%2Asqrt%285%29%29%2F%282%282%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+=+%282%281%2B3i%2Asqrt%285%29%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282%281%2B3i%2Asqrt%285%29%29%29%2F%282%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%281-3i%2Asqrt%285%29%29%29%2F%282%282%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282%281-3i%2Asqrt%285%29%29%29%2F%282%282%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+=+%281%2B3i%2Asqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%281%2B3i%2Asqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281-3i%2Asqrt%285%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%281-3i%2Asqrt%285%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We end up with the roots that were given to us. So this confirms that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the proper roots we were given. In other words, this confirms that the given roots lead to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: we could scale both sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by some nonzero number k, and get the same roots (since we can undo this and divide both sides by k). To keep things fairly simple, we'll make k = 1. We can consider <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be reduced as much as possible since the coefficients 2,-2,23 only have the common factor 1 between them. \r<BR>\n" );
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document.write( "Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2x%2B23+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2-2x%2B23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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