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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?

 Jan 18, 2022
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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?

 

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\(x^2 + y^2 + 2x - 4y + 8 - 10y + 3y\\ =x^2+2x+4+y^2-14y+49-45\\ =(x+2)^2+(y-7)^2-45\\ =(x+2)^2+(y-7)^2-45\\ \color{blue}x=-2\\ \color{blue}y=7\\ 0^2+0^2-45=\color{blue} -45 \)

 

The minimum value of the expression is - 45.

laugh  !

 Jan 18, 2022
edited by asinus  Jan 18, 2022

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