document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168783>Question 1168783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that if the sum of the sequence of the roots of the equation ax^2+bx+c=0 is 1 then b^2=2ac+a^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #793407 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=793407\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are p and q such that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"p+=+%28-b%2Bsqrt%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=q+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"q+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is from the quadratic formula\r<BR>\n" );
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document.write( "Adding p and q has the square root terms cancel out because we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28b%5E2-4ac%29 ALIGN=MIDDLE  ALT=\"sqrt%28b%5E2-4ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in both p and q; the only difference is that p has the positive version and q has the negative version. Effectively we're adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2B%28-R%29+=+0 ALIGN=MIDDLE  ALT=\"R%2B%28-R%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R+=+sqrt%28b%5E2-4ac%29 ALIGN=MIDDLE  ALT=\"R+=+sqrt%28b%5E2-4ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So after those root terms go away, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq+=+%28-b%2B%28-b%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"p%2Bq+=+%28-b%2B%28-b%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=p%2Bq+=+%28-2b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"p%2Bq+=+%28-2b%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=p%2Bq+=+-b%2Fa ALIGN=MIDDLE  ALT=\"p%2Bq+=+-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore, the sum of the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We're told that the roots add to 1, so we know further that,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq+=+1 ALIGN=MIDDLE  ALT=\"p%2Bq+=+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=-b%2Fa+=+1 ALIGN=MIDDLE  ALT=\"-b%2Fa+=+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=-b+=+a ALIGN=MIDDLE  ALT=\"-b+=+a\"   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=a+=+-b ALIGN=MIDDLE  ALT=\"a+=+-b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's plug that into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+2ac%2Ba%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+2ac%2Ba%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see what happens\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+2ac%2Ba%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+2ac%2Ba%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=b%5E2+=+2%28-b%29c%2B%28-b%29%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+2%28-b%29c%2B%28-b%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace 'a' with -b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+-2bc%2Bb%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+-2bc%2Bb%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=0+=+-2bc ALIGN=MIDDLE  ALT=\"0+=+-2bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract b^2 from both sides\r<BR>\n" );
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document.write( "From that last equation, we see that either b = 0 or c = 0.\r<BR>\n" );
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document.write( "If b = 0, then a = 0, but that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > isn't quadratic. Also, a = 0 causes division by zero errors in the quadratic formula. So we must make 'a' and b nonzero. This forces c to be zero.\r<BR>\n" );
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document.write( "Therefore, the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+2ac%2Ba%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+2ac%2Ba%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is only true if c = 0.\r<BR>\n" );
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document.write( "If we pick a nonzero c value such as c = 1, then,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+-2%2Bb%5E2 ALIGN=MIDDLE  ALT=\"b%5E2+=+-2%2Bb%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=0+=+-2 ALIGN=MIDDLE  ALT=\"0+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Which is a contradiction.<BR>\n" );
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