Answer:
Ok, the gravitational field in the x-axis can be written as:
g(x) = G*∑mₙ/(xₙ - x)
where the field points in the positive x-axis
where mₙ is the mass of the n-th particle, and xₙ is the position of the n-th particle, then, in our case we have:
g(x) = G*(- 2kg/x + 6kg(8m -x))
then; g(2) = G*( -2kg/2m + 6kg/(8m- 2m)) = G*( -1kg/m + 1kg/m) = 0
g(12) = G*( -(2/12)kg/m +6/(8 - 12)kg/m) = G* (-2/12 kg/m - 6/4kg/m) = -G*(20/12)kg/m
and we already find that the point where g(x) = 0 is 2
this is the x such:
G*(- 2kg/x + 6kg(8m -x)) = 0
then, the thing inside the parentheses must be zero, now we have:
2kg/x = 6kg(8m -x)
then:
x/2kg = (8/6) m/kg - x/(6kg)
x (1/(2kg) + 1/(6kg)) = (8/6)m/kg
x*(4/(6kg)) = (8/6)m/kg
x = (8/6)*(6/4)m = 2m
