What is the Natural Length of a Spring Based on Work Required for Stretching?

[chaotixmonjuish]

I'm having problems with the work section of my calc book. This is the problem:

If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch from 12 cm to 14 cm, what is the natural length of the spring?

I set up two equations:

6 J= Kx^2/2 evaluated at 10-L and 12-L
10 J= Kx^2/2 evaluated at 12-L and 14-L

I'm not sure what to do after the mess hell of algebra, I end up with a K and an L. Do I solve for K in one equation and plug it into the other?

 

[chaotixmonjuish]

I got a length of 8

 

[HallsofIvy]

If that's the easiest way to solve the two equations. Any any case, you want to eliminate K from the equations, resulting in single equation for L.