This resource is all about the Lagrange points and why they are important for the James Webb Space Telescope. Calculating the positions of these points is difficult but achievable with TI-Nspire.
The model is an inverse square with one ‘mass’ 100 times the other. Most accounts explaining the origins of the Lagrange points confuse gravitational potential with gravitational force. They also confuse Inertial reference frames with rotating reference frames and so talk about Coriolis forces and centrifugal forces as though they are real forces. School physics syllabuses usually insist that orbital motion is caused by a centripetal force, the force due to gravity and so use inertial reference frames. Force will be used rather than energy to find the position of the two points.
The approach is to consider the way in which the Earth’s gravitational field influences the Sun’s field by simply summing the two fields. Only the field on a straight line from the Sun through the Earth is considered.