document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=442903>Question 442903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you show a step by step process of how to put 25x^2+16y^2+150x=160y-225 into standard form for an ellipse? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #305507 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=305507\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2%2B16y%5E2%2B150x=160y-225 ALIGN=MIDDLE  ALT=\"25x%5E2%2B16y%5E2%2B150x=160y-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2%2B16y%5E2%2B150x-160y=-225 ALIGN=MIDDLE  ALT=\"25x%5E2%2B16y%5E2%2B150x-160y=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 160y from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x%5E2%2B150x%29%2B%2816y%5E2-160y%29=-225 ALIGN=MIDDLE  ALT=\"%2825x%5E2%2B150x%29%2B%2816y%5E2-160y%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%5E2%2B6x%29%2B%2816y%5E2-160y%29=-225 ALIGN=MIDDLE  ALT=\"25%28x%5E2%2B6x%29%2B%2816y%5E2-160y%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 25 from the first group (to make the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient equal to 1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%5E2%2B6x%29%2B16%28y%5E2-10y%29=-225 ALIGN=MIDDLE  ALT=\"25%28x%5E2%2B6x%29%2B16%28y%5E2-10y%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 16 from the second group (to make the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient equal to 1)\r<BR>\n" );
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document.write( "Take half of the \"x\" coefficient 6 to get 3. Square 3 to get 9. Add AND subtract this value inside the first parenthesis:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%5E2%2B6x%2B9-9%29%2B16%28y%5E2-10y%29=-225 ALIGN=MIDDLE  ALT=\"25%28x%5E2%2B6x%2B9-9%29%2B16%28y%5E2-10y%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add AND subtract 9 in the first parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%29=-225 ALIGN=MIDDLE  ALT=\"25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%2B9 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take half of the \"y\" coefficient -10 to get -5. Square -5 to get 25. Add AND subtract this value inside the second parenthesis:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%2B25-25%29=-225 ALIGN=MIDDLE  ALT=\"25%28%28x%2B3%29%5E2-9%29%2B16%28y%5E2-10y%2B25-25%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add AND subtract 25 in the second parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28%28x%2B3%29%5E2-9%29%2B16%28%28y-5%29%5E2-25%29=-225 ALIGN=MIDDLE  ALT=\"25%28%28x%2B3%29%5E2-9%29%2B16%28%28y-5%29%5E2-25%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2-10y%2B25 ALIGN=MIDDLE  ALT=\"y%5E2-10y%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-5%29%5E2 ALIGN=MIDDLE  ALT=\"%28y-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%2B3%29%5E2-25%289%29%2B16%28y-5%29%5E2-16%2825%29=-225 ALIGN=MIDDLE  ALT=\"25%28x%2B3%29%5E2-25%289%29%2B16%28y-5%29%5E2-16%2825%29=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%2B3%29%5E2-225%2B16%28y-5%29%5E2-400=-225 ALIGN=MIDDLE  ALT=\"25%28x%2B3%29%5E2-225%2B16%28y-5%29%5E2-400=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%2B3%29%5E2%2B16%28y-5%29%5E2-625=-225 ALIGN=MIDDLE  ALT=\"25%28x%2B3%29%5E2%2B16%28y-5%29%5E2-625=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=-225%2B625 ALIGN=MIDDLE  ALT=\"25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=-225%2B625\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 625 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=400 ALIGN=MIDDLE  ALT=\"25%28x%2B3%29%5E2%2B16%28y-5%29%5E2=400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825%28x%2B3%29%5E2%2B16%28y-5%29%5E2%29%2F400=cross%28400%2F400%29 ALIGN=MIDDLE  ALT=\"%2825%28x%2B3%29%5E2%2B16%28y-5%29%5E2%29%2F400=cross%28400%2F400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 400 (to make the right side equal to 1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825%28x%2B3%29%5E2%29%2F400%2B%2816%28y-5%29%5E2%29%2F400=1 ALIGN=MIDDLE  ALT=\"%2825%28x%2B3%29%5E2%29%2F400%2B%2816%28y-5%29%5E2%29%2F400=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B3%29%5E2%29%2F16%2B%28%28y-5%29%5E2%29%2F25=1 ALIGN=MIDDLE  ALT=\"%28%28x%2B3%29%5E2%29%2F16%2B%28%28y-5%29%5E2%29%2F25=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%2B3%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x%2B3%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite 16 as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2 ALIGN=MIDDLE  ALT=\"4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Rewrite 25 as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2 ALIGN=MIDDLE  ALT=\"5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-%28-3%29%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-%28-3%29%29%5E2%29%2F%284%5E2%29%2B%28%28y-5%29%5E2%29%2F%285%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-3%29 ALIGN=MIDDLE  ALT=\"x-%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now the equation is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the standard form of an ellipse where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-3 ALIGN=MIDDLE  ALT=\"h=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5 ALIGN=MIDDLE  ALT=\"k=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=4%5E2=16 ALIGN=MIDDLE  ALT=\"a%5E2=4%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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