Question 1209305
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Combine like terms to go from
{{{x^2 + y^2 - 4x + 2y + 3x^2 + 4y^2 - 12x + 15y + 24}}}
to
{{{4x^2 + 5y^2 - 16x + 17y + 24}}}


Let's complete the square for the x terms.
{{{4x^2 + 5y^2 - 16x + 17y + 24}}}
= {{{(4x^2-16x) + 5y^2 + 17y + 24}}}
= {{{4(x^2-4x) + 5y^2 + 17y + 24}}}
= {{{4(x^2-4x+red(0)) + 5y^2 + 17y + 24}}}
= {{{4(x^2-4x+red(4-4)) + 5y^2 + 17y + 24}}}
= {{{4((x^2-4x+4)-4) + 5y^2 + 17y + 24}}}
= {{{4((x-2)^2-4) + 5y^2 + 17y + 24}}}
= {{{4(x-2)^2 - 16 + 5y^2 + 17y + 24}}}
= {{{4(x-2)^2 + 5y^2 + 17y + 8}}}


Now complete the square for the y terms.
{{{4(x-2)^2 + 5y^2 + 17y + 8}}}
= {{{4(x-2)^2 + (5y^2 + 17y) + 8}}}
= {{{4(x-2)^2 + 5(y^2 + 3.4y) + 8}}}
= {{{4(x-2)^2 + 5(y^2 + 3.4y+red(0)) + 8}}}
= {{{4(x-2)^2 + 5(y^2 + 3.4y+red(2.89-2.89)) + 8}}}
= {{{4(x-2)^2 + 5((y^2 + 3.4y+2.89) - 2.89) + 8}}}
= {{{4(x-2)^2 + 5((y + 1.7)^2 - 2.89) + 8}}}
= {{{4(x-2)^2 + 5(y + 1.7)^2 +5(-2.89) + 8}}}
= {{{4(x-2)^2 + 5(y + 1.7)^2 -14.45 + 8}}}
= {{{4(x-2)^2 + 5(y + 1.7)^2 - 6.45}}}
Each decimal value mentioned is exact and hasn't been rounded.


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After completing the square for both x and y we go from
{{{4x^2 + 5y^2 - 16x + 17y + 24}}}
to
{{{4(x-2)^2 + 5(y + 1.7)^2 - 6.45}}}
You can verify this by expanding everything out in that 2nd expression, then simplifying, to arrive back at the 1st expression.
Another way to verify is to use something like <a 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>


The smallest that the portion {{{(x-2)^2}}} can get is 0 and the same goes for the portion {{{(y+1.7)^2}}}
Think of the parabola y = x^2


Therefore the coldest temperature is
{{{4*red((x-2)^2) + 5*red((y + 1.7)^2) - 6.45 = 4*red(0) + 5*red(0) - 6.45 = -6.45}}}
Whether it is Celsius or Fahrenheit, it's not clear. 


Side note: The location of this coldest point is (x,y) = (2, -1.7) since this x,y pairing makes {{{(x-2)^2=0}}} and {{{(y+1.7)^2 = 0}}} true.
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. 


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Answer = <font color=red>-6.45</font>
This value is exact. It hasn't been rounded.
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