document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074872>Question 1074872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of values of a for which [(a^2)-3a+4](x^2) + [(a^2)-5a+6]x  + [(a^2)-4]=0<BR>\n" );
document.write( " is an identity in x is <BR>\n" );
document.write( "i)   0<BR>\n" );
document.write( "ii)  2<BR>\n" );
document.write( "iii) 1<BR>\n" );
document.write( "iv) 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689710 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=689710\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If that is an identities, it should be true for all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we need to have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-4=0 ALIGN=MIDDLE  ALT=\"a%5E2-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=4 ALIGN=MIDDLE  ALT=\"a%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are values of a that comply with that, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2 ALIGN=MIDDLE  ALT=\"a=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but would those values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > work with any and all values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( " <BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=4 ALIGN=MIDDLE  ALT=\"a%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <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;\" > we need to have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-3a%2B4%29%2B%284-5a%2B6%29%2B0=0 ALIGN=MIDDLE  ALT=\"%284-3a%2B4%29%2B%284-5a%2B6%29%2B0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-3a%2B4%2B4-5a%2B6=0 ALIGN=MIDDLE  ALT=\"4-3a%2B4%2B4-5a%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and further simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18-8a=0 ALIGN=MIDDLE  ALT=\"18-8a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is not true for either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2 ALIGN=MIDDLE  ALT=\"a=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280%29 ALIGN=MIDDLE  ALT=\"highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > that make that expression an identity.<BR>\n" );
document.write( "We could not find even one that would work for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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>\n" );
document.write( "so none would work for every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER WAY:<BR>\n" );
document.write( "The only way that <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;\" > would be an identity in x<BR>\n" );
document.write( "is if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b=c=0 ALIGN=MIDDLE  ALT=\"a=b=c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the only way that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%28a%29%2Ax%5E2%2BG9a%29%2Ax%2BH%28a%29=0 ALIGN=MIDDLE  ALT=\"F%28a%29%2Ax%5E2%2BG9a%29%2Ax%2BH%28a%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "could be an identity in x for some value of a is if<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28F%28a%29=0%2CG%28a%29=0%2CH%28a%29=0%29 ALIGN=MIDDLE  ALT=\"system%28F%28a%29=0%2CG%28a%29=0%2CH%28a%29=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has some solution.<BR>\n" );
document.write( "We could have tried to find the solutions to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-3a%2B4=0 ALIGN=MIDDLE  ALT=\"a%5E2-3a%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (no solution),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-5a%2B6=0 ALIGN=MIDDLE  ALT=\"a%5E2-5a%2B6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a=3 ans a=2) , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-4=0 ALIGN=MIDDLE  ALT=\"a%5E2-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a=2 and a=-2),<BR>\n" );
document.write( "and then concluded that no solution worked for all 3 equations.<BR>\n" );
document.write( "That may be inefficient, but it may be what was intended,<BR>\n" );
document.write( "to make practice solving quadratic equations.<BR>\n" );
document.write( "(Of course, if you start by finding that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-3a%2B4=0 ALIGN=MIDDLE  ALT=\"a%5E2-3a%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has no real solution,<BR>\n" );
document.write( "you do not need to work on solving teh other equations </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );