Given z=x²+3xy-2y²

z(2,3)=4+18-18 =4
z(2.5,2.5)=6.25 +18.75-12.5 =12.5
Change in z is 8.5

(Its a long time since I did partial differentiation which may, for all I know, give a different result)

I see from the help given elsewhere that the "estimate" is 9.5.

Use the linear approximateion formula Δz ≈ (∂z/∂x) Δx + (∂z/∂y) Δy, where the partials are evaluated at (x,y) = (2,3).

∂z/∂x = 2x + 3y = 13
∂z/∂y = 3x - 4y = -6
Δx = 0,5
Δy = -0,5
Δz = 6,5 + 3 = 9,5

So 9,5 is indeed right.