+1248 Find the minimum value of the expression $x^2+y^2+2x-4y+8+10x-12y$ for real $x$ and $y$.
+36943 = x^2 +12x + y^2 - 16y = -8 complete the square for 'x' and 'y'
(x+6)^2 + (y-8)^2 = -8 + 36 + 64
(x+6)^2 + (y-8)^2 = 92 <======= this is a circle centered at (-6, 8) with radius sqrt (92)
the minimum point will be sqrt (92) below the center : ( -6, 8-sqrt(92) )
so the minumum VALUE would be 8 - sqrt (92)
Here is a picture:
+36943 = x^2 +12x + y^2 - 16y = -8 complete the square for 'x' and 'y'
(x+6)^2 + (y-8)^2 = -8 + 36 + 64
(x+6)^2 + (y-8)^2 = 92 <======= this is a circle centered at (-6, 8) with radius sqrt (92)
the minimum point will be sqrt (92) below the center : ( -6, 8-sqrt(92) )
so the minumum VALUE would be 8 - sqrt (92)
Here is a picture:
