document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139198>Question 1139198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find nonzero integers A, B, C such that Ax^2+Bx+C=0 has solutions B and C </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756981 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=756981\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "In the quadratic equation<br><BR>\n" );
document.write( "<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;\" ><br><BR>\n" );
document.write( "the sum of the roots is -B/A and the product of the roots is C/A.<br><BR>\n" );
document.write( "If the roots are B and C, then<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2BC+=+-B%2FA ALIGN=MIDDLE  ALT=\"B%2BC+=+-B%2FA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC+=+C%2FA ALIGN=MIDDLE  ALT=\"BC+=+C%2FA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Equation (2) gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC+=+C ALIGN=MIDDLE  ALT=\"ABC+=+C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB+=+1 ALIGN=MIDDLE  ALT=\"AB+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There are two possibilities with A and B both integers: they are both 1, or they are both -1.<br><BR>\n" );
document.write( "If A = B = 1 then equation (1) gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2BC+=+-1 ALIGN=MIDDLE  ALT=\"1%2BC+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+-2 ALIGN=MIDDLE  ALT=\"C+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then the equation is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x-1%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and the roots are B and C.<br><BR>\n" );
document.write( "So there is one solution to the problem.<br><BR>\n" );
document.write( "If A = B = -1, then equation (1) gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2BC+=+-1 ALIGN=MIDDLE  ALT=\"-1%2BC+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+0 ALIGN=MIDDLE  ALT=\"C+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the requirement is that A, B, and C be non-zero, there is no solution in this case.<br><BR>\n" );
document.write( "So the unique quadratic equation Ax^2+Bx+C=0 with roots B and C is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );