Simplifying equations (thought process)

[jongro]

Take this equation:
f(x) = u^0.5(2 - 2u) + (2u - u²)(0.5u^-0.5)

My lecturer simplified it to this:
f(x) = (2u(2 - 2u) + (2u - u²)) / (2u^0.5)
= (6u - 5u²) / (2u^0.5)

My intuition tells me to simplify it like this:
f(x) = 2u^0.5 - 2u^1.5 + u^0.5 - 0.5u^1.5
= sqrt(u)(3 - (5/2)u)

My problem is that, while I can see how they're both valid, I cannot understand how in the world how the lecturer could think like that?

This question probably sounds ridiculous, but could someone please explain to me the thought process involved in simplifying the equation like the lecturer did? It just seems REALLY unnatural for me.

What are some of the things that mathematicians look for when they have an equation that they want to factorize and simplify?

I think that there is something really messed up with the way I think about equations.

 

[symbolipoint]

Distributive property for the first expression and transformation of the second expression into a form with only positive exponent. Next process was simple fraction arithmetic.

 

[Gerenuk]

What are some of the things that mathematicians look for when they have an equation that they want to factorize and simplify?
I think that there is something really messed up with the way I think about equations.

To be honest I don't believe your lecturer is very confident at manipulating equations. He probably did the first thing that came into his mind instead of thinking twice and doing the better way. I think your way is best.