document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252215>Question 252215</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all real values of x for which the functionsare equal. Give your answers to the nearest hundredth.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "p(x)=x^2-2x+1,Q(x)=-x^3+3x^2-3x+3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "P(x)=x^4-5x^2+4,Q(x)=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184075 by </B><A HREF=/tutors/aboutme.mpl?userid=drk><B>drk(1908)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drk><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=184075\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's look at the first equation set. \r<BR>\n" );
document.write( "\n" );
document.write( "x^2 -2X+1 = -X^3+3x^2 - 3X + 3<BR>\n" );
document.write( "X^3 -2X^2 + X -2 = 0<BR>\n" );
document.write( "x^2(X-2) + 1(X-2) = 0<BR>\n" );
document.write( "(X^2+1)(X-2) = 0<BR>\n" );
document.write( "So, we set each part = 0.<BR>\n" );
document.write( "X^2 + 1 = 0<BR>\n" );
document.write( "X^2 = -1. Here we have either no solution or +-i<BR>\n" );
document.write( "X-2 = 0.<BR>\n" );
document.write( "X=2. This is the only real solution.\r<BR>\n" );
document.write( "\n" );
document.write( "Now, let's look at the second equation set.<BR>\n" );
document.write( "X^4 - 5X^2 +4 = 2<BR>\n" );
document.write( "X^4 - 5X^2 +2 = 0<BR>\n" );
document.write( "Let U = X^2. This a \"dummy variable\" substitution. We get<BR>\n" );
document.write( "U^2 - 5U +2 = 0.<BR>\n" );
document.write( "We apply the quadratic to get<BR>\n" );
document.write( "U = (5 +- sqrt(25-8))/2<BR>\n" );
document.write( "U = (5 +- sqrt(17))/2.<BR>\n" );
document.write( "We resubstitute X^2 for U and get<BR>\n" );
document.write( "X^2 = (5 +- sqrt(17))/2.<BR>\n" );
document.write( "X = +-sqrt((5 +-sqrt(17))/2). These are both real solutions.  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );