document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=916327>Question 916327</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The distance d, to the horizon of an object h miles above a planet's surface is given by the equation {d= sqrt(6319h+h^2)}. How many miles above the planets surface is a satellite if the distance to the horizon is 720 mi? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #556053 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=556053\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> All that is expected is that you plug the value for  the distance to the horizon <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=720 ALIGN=MIDDLE  ALT=\"d=720\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=+sqrt%286319h%2Bh%5E2%29 ALIGN=MIDDLE  ALT=\"d=+sqrt%286319h%2Bh%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and then solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720=sqrt%286319h%2Bh%5E2%29 ALIGN=MIDDLE  ALT=\"720=sqrt%286319h%2Bh%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The solutions to that equation will also be solutions to the equation we get by squaring both sides of the equal sign.<BR>\n" );
document.write( "(The new equation may have some other extra solutions, but we will worry about that later).<BR>\n" );
document.write( "Squaring both sides of the equal sign, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=720%5E2=6319h%2Bh%5E2 ALIGN=MIDDLE  ALT=\"720%5E2=6319h%2Bh%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=518400=6319h%2Bh%5E2 ALIGN=MIDDLE  ALT=\"518400=6319h%2Bh%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2%2B6319h-518400=0 ALIGN=MIDDLE  ALT=\"h%5E2%2B6319h-518400=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That quadratic equation can be solved by factoring,<BR>\n" );
document.write( "if you realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6400-81=6319 ALIGN=MIDDLE  ALT=\"6400-81=6319\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6400%2A%28-81%29=-518400 ALIGN=MIDDLE  ALT=\"6400%2A%28-81%29=-518400\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then you would know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2%2B6319h-518400=%28x-81%29%28x%2B6400%29 ALIGN=MIDDLE  ALT=\"h%5E2%2B6319h-518400=%28x-81%29%28x%2B6400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and you would look for the solutions to the equivalent equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-81%29%28x%2B6400%29=0 ALIGN=MIDDLE  ALT=\"%28x-81%29%28x%2B6400%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=81%29 ALIGN=MIDDLE  ALT=\"highlight%28x=81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6400 ALIGN=MIDDLE  ALT=\"x=-6400\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the satellite is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2881%29 ALIGN=MIDDLE  ALT=\"highlight%2881%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles above the planet's surface.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Of course, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6400 ALIGN=MIDDLE  ALT=\"x=-6400\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an extraneous solution,<BR>\n" );
document.write( "meaning that is not a solution of the original equation,<BR>\n" );
document.write( "but it is a solution we \"gained\" when we squared both sides of the equal sign.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you do not solve by factoring, you can always get the same results by applying the quadratic formula (or by completing he square).\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
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