document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=694263>Question 694263</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you design a nonlinear system that has at least two solutions and one solution must be the ordered pair, (-2, 5)? Can you please tell me how you came up with your system and give the entire solution set for the system. Thank you in advance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427849 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427849\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are infinite solutions to this problem, however the following should suffice.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Assume you have a circle that contains the given point, (-2,5).  Further assume that the center of this circle is the origin.  Using the distance formula, we find the radius of this circle to be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\ =\ \sqrt{(-2\ -\ 0)^2\ +\ (5\ -\ 0)^2}\ =\ \sqrt{29}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then using the pattern for the equation of a circle of radius <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\"> centered at the origin, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2\ +\ y^2\ =\ r^2\">, we write the equation of the required circle:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ y^2\ =\ 29\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now assume a parabola with a vertex on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-axis and that contains the point (-2,5).  By symmetry, such a parabola must also pass through the point (2,5).  I leave it as an exercise for the student to convince himself of the fact that this other point is, indeed, on both the parabola and the circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Just for the sake of simplicity of arithmetic, let's choose (0,-3) as the vertex for the parabola.  In fact, any point on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-axis would do, but this particular point results in tidy arithmetic.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now that we have three non-collinear points, we can write the equation of a parabola that contains all three points.  Note that the general form of a parabola is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the particular parabola for which we want an equation must contain the point (-2,5).  That is to say, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ -2\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ 5\">.  In other words,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(-2)^2\ +\ b(-2)\ +\ c\ =\ 5\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "or more simply put:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a\ -\ 2b\ +\ c\ =\ 5\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Similarly, since the parabola must contain the points (2, 5) and (0, -3),\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4a\ +\ 2b\ +\ c\ =\ 5\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "and\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0a\ +\ 0b\ +\ c\ =\ -3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is to say\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c\ =\ -3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solving this system of equations to obtain the coefficients for the parabola equation is left as an exercise for the student.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Obviously, the solution set is the two points that the circle and the parabola have in common, namely (-2,5) and (2,5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Super Double Plus Extra Credit: Find a point for the vertex of the parabola that results in 3 points in the solution set of the system.  Find another point for the vertex that results in 4 points in the solution set.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );