Why do I see some sources define the work on a spring as W = (1/2)(k)(xf^2 - xi^2) and some sources define it as W = (1/2)(k)(x^2) ?

For example (I'll test out both equations):

Let xi = 3, xf = 7, and k = 3
So by using W = (1/2)(k)(xf^2 - xi^2)
= (1/2)(3)(7^2 - 3^2)
= (1/2)(3)(49 - 9)
= (1/2)(3)(40)
= 60

By using W = (1/2)(k)(x^2)
= (1/2)(k)(xf - xi)^2
= (1/2)(3)(7-3)^2
= (1/2)(3)(4)^2
= (1/2)(3)(16)
= 24

I'm confused since I didn't get the same result for both equations, so they are not the same then? Am I miscalculating something?

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  1. In the second equation, it assumes the initial x is zero.

    Work= INtegral force(x)*dx= INT k*x*dx over limits or
    work= 1/2 k (xf^2-xi^2)

  2. Oh I think I get it now. So if xi = 0, then both versions of the equation will give the same result. Otherwise, if the initial position is not zero, then W = (1/2)(k)(xf^2 - xi^2) MUST be used. Correct?

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  3. correct. The second equation is for beginners (xi=0)

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