Least value of expression x^2+4y^2+3z^2-4x-20y-12z+45 is.?

Hello,

4 is the least value.

Indeed:
x²-4x+4=(x-2)²
4y²-20y+25=(2y-5)²
3(z²-4z+4)=3(z-2)²

x²+4y²+3z²-4x-20y-12z+45= (x-2)²+(2y-5)²+3(z-2)²+45-4-25-12
=(x-2)²+(2y-5)²+3(z-2)²+4

Minimum is 4 when  x=2,y=5/2 and z=2 for square is always positive.

Idon't see any graph, however the graph of the function : f(x) = x²-2x+3 is a parabola: 1) open upward (since a> 0)2) it intersects y-axis in 3 (y=3)3) its axis of symmetry is x = +1 ( -b/2a)4) it's minimum is y = -2, so its vertex is (1,-2)5) it doesn't intersect the x-axis since discriminant < 0