document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=408299>Question 408299</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given one roots of x^2+px+8=0 is two times of the other. Find the roots and p. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287703 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=287703\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Interesting thing about this problem is that the roots and the value of p are not uniquely determined. You'll see why:\r<BR>\n" );
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document.write( "Suppose the two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r ALIGN=MIDDLE  ALT=\"2r\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The product of the roots is 8 by Vieta's formulas, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r%5E2+=+8 ALIGN=MIDDLE  ALT=\"2r%5E2+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2+=+4 ALIGN=MIDDLE  ALT=\"r%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> r = +/- 2. Hence, the roots are {2, 4} or {-2, -4}. Also, the sum of the roots is -p, so p = -6 or 6, respectively.\r<BR>\n" );
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document.write( "Hence, the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+6x+%2B+8 ALIGN=MIDDLE  ALT=\"x%5E2+-+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > has roots {2, 4} and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > has roots {-2, -4}. </SPAN>\n" );
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