document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=189361>Question 189361</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Find the standard form of the hyperbola given by the equation   4x^2 - 25y^2 - 50y - 125= 0 \r<BR>\n" );
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document.write( "Please help! <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142056 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=142056\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4x^2 - 25y^2 - 50y - 125= 0\">\r<BR>\n" );
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document.write( "And you want to obtain:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{(x - h)^2}{a^2} - \frac{ (y - k)^2}{b^2} = 1\">\r<BR>\n" );
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document.write( "Where (<i>h</i>,<i>k</i>) is the point about which the hyperbola is centered, <i>a</i> is the semi-major axis, and <i>b</i> is the semi-minor axis.\r<BR>\n" );
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document.write( "The process involves completing the square separately for the <i>x</i> and <i>y</i> variables.\r<BR>\n" );
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document.write( "First, put the constant term on the right:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4x^2 - 25y^2 - 50y = 125\">\r<BR>\n" );
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document.write( "Since there is no 1st degree <i>x</i> term, you can factor directly:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4(x - 0)^2 - 25y^2 - 50y = 125\">\r<BR>\n" );
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document.write( "Take the lead coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> term and factor it out of the <i>y</i> terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4(x - 0)^2 - 25 (y^2 + 2y +\ \ \ ) = 125\">\r<BR>\n" );
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document.write( "Divide the coefficient on the 1st order <i>y</i> term, square the result, and add to both sides.  Remember, since you have the factor of -25 outside of the parentheses, you are actually adding -25 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4(x - 0)^2 - 25 (y^2 + 2y + 1 ) = 125 - 25\">\r<BR>\n" );
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document.write( "Factor the perfect square inside of the parentheses:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  4(x - 0)^2 - 25 (y + 1)^2 = 100\">\r<BR>\n" );
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document.write( "Divide by the constant term on the right so that the right hand side becomes 1:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{4(x - 0)^2}{100} - \frac{25 (y + 1)^2}{100} = 1\">\r<BR>\n" );
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document.write( "Reduce the fractions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{(x - 0)^2}{25} - \frac{ (y + 1)^2}{4} = 1\">\r<BR>\n" );
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document.write( "Express the denominators as the square root squared:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \frac{(x - 0)^2}{5^2} - \frac{ (y + 1)^2}{2^2} = 1\">\r<BR>\n" );
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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" );
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