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I have been trying to find the smallest possible value of $x^2 + 4xy + 5y^2 - 4x - 6y + 7$, but I do not seem to have been heading in any direction which is going to give me an answer I feel certain is correct. Any hints on how to algebraically approach finding this value would be appreciated. I prefer not to be told what the value is.

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You could take derivatives and set them to $0$, so solving $2x+4y-4=0$ and $4x+10y-6=0$ will give $x=4$ and $y=-1$. This will be a minimum, maximum or saddle point of the function. –  Henry Jun 21 '12 at 6:57

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Note that $x^2 + 4xy + 5y^2 - 4x - 6y + 7=(x+2y)^2+y^2-4x-6y+7$.

Let $u=x+2y$. Write our expression in terms of $u$ and $y$, and complete the squares.

Remark: The approach may have looked a little ad hoc, but the same basic idea will work for any quadratic polynomial $Q(x,y)$ that has a minimum or maximum. For quadratic polynomials $Q(x_1,x_2,\dots, x_n)$, one can do something similar, but it becomes useful to have more theory. You may want to look into the general diagonalization procedure.

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$$x^2 + 4xy + 5y^2 -4x -6y + 7 = (x+2y-2)^2 + (y+1)^2 + 2$$

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