document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209305>Question 1209305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The temperature of a point (x,y) in the plane is given by the expression x^2 + y^2 - 4x + 2y + 3x^2 + 4y^2 - 12x + 15y + 24.  What is the temperature of the coldest point in the plane? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848183 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=848183\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Combine like terms to go from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+-+4x+%2B+2y+%2B+3x%5E2+%2B+4y%5E2+-+12x+%2B+15y+%2B+24 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+-+4x+%2B+2y+%2B+3x%5E2+%2B+4y%5E2+-+12x+%2B+15y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Let's complete the square for the x terms.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2-16x%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"%284x%5E2-16x%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2-4x%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28x%5E2-4x%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2-4x%2Bred%280%29%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28x%5E2-4x%2Bred%280%29%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2-4x%2Bred%284-4%29%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28x%5E2-4x%2Bred%284-4%29%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28%28x%5E2-4x%2B4%29-4%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28%28x%5E2-4x%2B4%29-4%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28%28x-2%29%5E2-4%29+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28%28x-2%29%5E2-4%29+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+-+16+%2B+5y%5E2+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+-+16+%2B+5y%5E2+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5y%5E2+%2B+17y+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5y%5E2+%2B+17y+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Now complete the square for the y terms.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5y%5E2+%2B+17y+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5y%5E2+%2B+17y+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+%285y%5E2+%2B+17y%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+%285y%5E2+%2B+17y%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%2Bred%280%29%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%2Bred%280%29%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%2Bred%282.89-2.89%29%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y%5E2+%2B+3.4y%2Bred%282.89-2.89%29%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28%28y%5E2+%2B+3.4y%2B2.89%29+-+2.89%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28%28y%5E2+%2B+3.4y%2B2.89%29+-+2.89%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28%28y+%2B+1.7%29%5E2+-+2.89%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28%28y+%2B+1.7%29%5E2+-+2.89%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+%2B5%28-2.89%29+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+%2B5%28-2.89%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-14.45+%2B+8 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-14.45+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-+6.45 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-+6.45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each decimal value mentioned is exact and hasn't been rounded.\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "\n" );
document.write( "After completing the square for both x and y we go from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-+6.45 ALIGN=MIDDLE  ALT=\"4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-+6.45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can verify this by expanding everything out in that 2nd expression, then simplifying, to arrive back at the 1st expression.<BR>\n" );
document.write( "Another way to verify is to use something like <a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=4x%5E2+%2B+5y%5E2+-+16x+%2B+17y+%2B+24%3D4%28x-2%29%5E2+%2B+5%28y+%2B+1.7%29%5E2+-+6.45\">WolframAlpha</a>\r<BR>\n" );
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document.write( "\n" );
document.write( "The smallest that the portion <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can get is 0 and the same goes for the portion <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B1.7%29%5E2 ALIGN=MIDDLE  ALT=\"%28y%2B1.7%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Think of the parabola y = x^2\r<BR>\n" );
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document.write( "Therefore the coldest temperature is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ared%28%28x-2%29%5E2%29+%2B+5%2Ared%28%28y+%2B+1.7%29%5E2%29+-+6.45+=+4%2Ared%280%29+%2B+5%2Ared%280%29+-+6.45+=+-6.45 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Whether it is Celsius or Fahrenheit, it's not clear. \r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Side note: The location of this coldest point is (x,y) = (2, -1.7) since this x,y pairing makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2B1.7%29%5E2+=+0 ALIGN=MIDDLE  ALT=\"%28y%2B1.7%29%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > true.<BR>\n" );
document.write( "If you plugged x = 2 and y = -1.7 into 4x^2 + 5y^2 - 16x + 17y + 24, or the original expression if you wanted, you should get -6.45 as the result. \r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Answer = <font color=red>-6.45</font><BR>\n" );
document.write( "This value is exact. It hasn't been rounded.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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