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What is the minimum value of the expression x^2 + y^2 + 2x - 4y + 8 - 10y + 3y for real x and y?

 May 27, 2021
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$x^2+2x+y^2-11y+8=(x+1)^2+\left(y-\frac{11}{2}\right)^2+8-1-\frac{11^2}{2^2}\ge 8-1-\frac{121}{4}=-\frac{93}{4}$. 

 May 27, 2021

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