Is there a way to find the lowest possible value of the following expression:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] x^2 + 4xy + 5y^2 - 4x - 6y +7 [/tex] without using calculus?

I've just got myself a different textbook and am working from the beginning, trying to use only the tools given to me thus far in the book (within reason).

One way to solve it is to complete the squares for x and y to get

[itex] (x-2)^2+5(y- \frac{3}{5})^2 +4xy + \frac {6}{5} [/itex] and then find the value of 4xy which keeps the sum of both squares and itself as small as possible..ie differentiate

Problem is the book has not not even touched upon calculus yet and so I suspect there is a more elementary approach that I have either missed or am not even aware of and this is bugging me :grumpy: ...The furthest I got was the factorisation above and figuring out that 4xy should be negative and its absolute value being as high as possible so that y < 0 whilst x > 2 ... but I still cannot find a basic method of finding the precise values of x and y. Anyone have any suggestions?

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# Homework Help: Factorisation with 2 variables

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