document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209661>Question 1209661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the conic section represented by the equation\r<BR>\n" );
document.write( "\n" );
document.write( "3x^2 + y^2 + 9x - 5y - 20 = 8x^2 + 6x + 47\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849777 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849777\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to identify the conic section:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Rearrange the equation:**  Move all terms to one side:\r<BR>\n" );
document.write( "\n" );
document.write( "   3x² + y² + 9x - 5y - 20 - 8x² - 6x - 47 = 0<BR>\n" );
document.write( "   -5x² + 3x + y² - 5y - 67 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Multiply by -1 (optional, but makes some calculations easier):**\r<BR>\n" );
document.write( "\n" );
document.write( "   5x² - 3x - y² + 5y + 67 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Complete the square for both x and y:**\r<BR>\n" );
document.write( "\n" );
document.write( "   5(x² - (3/5)x) - (y² - 5y) + 67 = 0<BR>\n" );
document.write( "   5(x² - (3/5)x + (3/10)²) - 5(3/10)² - (y² - 5y + (5/2)²) + (5/2)² + 67 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - 5(9/100) - (y - 5/2)² + 25/4 + 67 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - 9/20 - (y - 5/2)² + 25/4 + 67 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - (y - 5/2)² + 67 - 9/20 + 125/20 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - (y - 5/2)² + 67 + 116/20 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - (y - 5/2)² + 67 + 5.8 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - (y - 5/2)² + 72.8 = 0<BR>\n" );
document.write( "   5(x - 3/10)² - (y - 5/2)² = -72.8\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Divide by -72.8:**\r<BR>\n" );
document.write( "\n" );
document.write( "   [5(x - 3/10)²] / -72.8 - [(y - 5/2)²] / -72.8 = 1<BR>\n" );
document.write( "   [(x - 3/10)²] / (-72.8/5) - [(y - 5/2)²] / -72.8 = 1<BR>\n" );
document.write( "   [(x - 3/10)²] / -14.56 - [(y - 5/2)²] / -72.8 = 1\r<BR>\n" );
document.write( "\n" );
document.write( "5. **Analyze the equation:**\r<BR>\n" );
document.write( "\n" );
document.write( "   The equation is in the form:\r<BR>\n" );
document.write( "\n" );
document.write( "   (x²/a²) - (y²/b²) = 1  (Hyperbola)<BR>\n" );
document.write( "   OR<BR>\n" );
document.write( "   -(x²/a²) + (y²/b²) = 1 (Hyperbola)\r<BR>\n" );
document.write( "\n" );
document.write( "Since both the x² and y² terms are present and have *opposite* signs, the equation represents a **hyperbola**.  The negative signs simply determine the orientation of the hyperbola (whether it opens horizontally or vertically). Since the x^2 term is negative, this is a hyperbola that opens up and down.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );