document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199157>Question 1199157</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the real solutions of the equation ax^2 + bx + c = 0 are the reciprocation the real solutions of the equation cx^2 + bx + a = 0. Assume that b^2 - 4ac is greater than or equal to 0.\r<BR>\n" );
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document.write( "Can someone get me started here?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832859 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=832859\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Show that the real solutions of the equation ax^2 + bx + c = 0 are the reciprocation <BR>\n" );
document.write( "the real solutions of the equation cx^2 + bx + a = 0. <BR>\n" );
document.write( "Assume that b^2 - 4ac is greater than or equal to 0.<BR>\n" );
document.write( "Can someone get me started here?<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As given, &nbsp;the problem's formulation is not &nbsp;100% &nbsp;accurate.<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To be accurate, &nbsp;it must assume that the coefficient &nbsp;\"c\" &nbsp;is not zero,<BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;because otherwise an equation &nbsp;cx^2 + bx + a = 0 &nbsp;IS &nbsp;NOT a &nbsp;QUADRATIC.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This assumption is equivalent to say that no one of the routes of <BR>\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the original equation is &nbsp;0 &nbsp;(zero).\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is a &nbsp;NECESSARY &nbsp;condition to that the reciprocal to the roots do exist.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;in my solution below &nbsp;I &nbsp;will assume that &nbsp;a=/=0, &nbsp;c=/=0.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Let x be a root of the equation ax^2 + bx + c = 0.\r\n" );
document.write( "Since c=/= 0, it implies that x=/=0, so the reciprocal  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >  does exist.\r\n" );
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document.write( "Divide equation  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0 by  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  You will get\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b%2A%281%2Fx%29+%2B+c%2A%281%2Fx%29%5E2 ALIGN=MIDDLE  ALT=\"a+%2B+b%2A%281%2Fx%29+%2B+c%2A%281%2Fx%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0.\r\n" );
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document.write( "It means that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx ALIGN=MIDDLE  ALT=\"1%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >  is the root of the equation  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+bx+%2B+cx%5E2 ALIGN=MIDDLE  ALT=\"a+%2B+bx+%2B+cx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 0.\r\n" );
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document.write( "Q.E.D.\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "By the way, the statement of the problem is true for complex roots, too,\r<BR>\n" );
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document.write( "as it is true for real roots.\r<BR>\n" );
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document.write( "The same formal proof works for the complex root.\r<BR>\n" );
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document.write( "So, the requirement of the problem to the roots to be real is not necessary: it is EXCESSIVE.\r<BR>\n" );
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