document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209307>Question 1209307</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x and y be nonnegative real numbers.  If x^2 + 5y^2 = 30, then find the maximum value of x^2 + y^2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848212 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848212\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll refer to the two nonnegative numbers as m and n.\r<BR>\n" );
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document.write( "m^2 + 5n^2 = 30 rearranges to m^2 = 30 - 5n^2\r<BR>\n" );
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document.write( "Plug that into the expression we want to max out.<BR>\n" );
document.write( "m^2 + n^2<BR>\n" );
document.write( "= 30-5n^2 + n^2<BR>\n" );
document.write( "= 30-4n^2\r<BR>\n" );
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document.write( "We want to make 30-4n^2 as large as possible.<BR>\n" );
document.write( "This is equivalent to looking for the highest point on the upside down parabola y = 30-4x^2 aka y = -4x^2+30.\r<BR>\n" );
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document.write( "The smallest that x^2 can get is 0.<BR>\n" );
document.write( "The negative out front will flip things to show that -4x^2 maxes out when x = 0<BR>\n" );
document.write( "Therefore y = -4x^2+30 maxes out when x = 0 and its paired value is y = 30.\r<BR>\n" );
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document.write( "The vertex of y = -4x^2+30 is (0,30) which is the highest point on this parabola.<BR>\n" );
document.write( "You can use a graphing tool such as <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> and <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> to confirm. <BR>\n" );
document.write( "Or you can go old-school with something like a TI83.\r<BR>\n" );
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document.write( "The y coordinate of this vertex is the max value of 30-4n^2 aka m^2 + n^2 when m^2+5n^2 = 30.<BR>\n" );
document.write( "A somewhat similar question is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Quadratic-relations-and-conic-sections.faq.question.1208800.html\">here</a>\r<BR>\n" );
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document.write( "Answer: <font color=red>30</font><BR>\n" );
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