Difficulty in deciding when to apply work energy theorem

[physicsissohard]

Homework Statement

Two blocks A and B of the same mass connected with a spring are placed on a rough inclined plane, which makes an angle $\theta$ with horizontal. What minimum velocity should be given to A up the incline so that B just moves

Relevant Equations

its

This is how I tried to do it. The force required to move B up the incline is $kx$ where x is elongation and k is spring constant. we know that spring force is greater than $mg(sin\theta+\mu cos\theta)$. And we can use work-energy theorem to figure out velocity.
$0.5*k*x^2=0.5*mv^2$ where $0.5*k*x^2$ is work done by spring force. and when you count all the chickens $v$ turns out to be $\sqrt{km}(gsin\theta+\mu gcos\theta)$. Which apparently is the wrong answer. And the correct answer apparently is $$\sqrt{(3m)/k}(gsin\theta+\mu gcos\theta)$$. I have no idea what I did wrong. Can somebody help? is there something wrong with the WOrk energy theorem, or what?
[![enter image description here][1]][1] [1]: https://i.stack.imgur.com/81yAK.png

 

[nasu]

The NET work is the one equal to the change in kinetic energy. This is what the work energy theorem "says".

 

[PeroK]

Homework Statement: Two blocks A and B of the same mass connected with a spring are placed on a rough inclined plane, which makes an angle $\theta$ with horizontal. What minimum velocity should be given to A up the incline so that B just moves
Relevant Equations: its

This is how I tried to do it. The force required to move B up the incline is $kx$ where x is elongation and k is spring constant. we know that spring force is greater than $mg(sin\theta+\mu cos\theta)$. And we can use work-energy theorem to figure out velocity.
$0.5*k*x^2=0.5*mv^2$ where $0.5*k*x^2$ is work done by spring force. and when you count all the chickens $v$ turns out to be $\sqrt{km}(gsin\theta+\mu gcos\theta)$. Which apparently is the wrong answer. And the correct answer apparently is $$\sqrt{(3m)/k}(gsin\theta+\mu gcos\theta)$$. I have no idea what I did wrong. Can somebody help? is there something wrong with the WOrk energy theorem, or what?
[![enter image description here][1]][1] [1]: https://i.stack.imgur.com/81yAK.png

You need two hash signs for your inline Latex.

Is the spring the only force on block A?