Take this equation: f(x) = u0.5(2 - 2u) + (2u - u2)(0.5u-0.5) My lecturer simplified it to this: f(x) = (2u(2 - 2u) + (2u - u2)) / (2u0.5) = (6u - 5u2) / (2u0.5) My intuition tells me to simplify it like this: f(x) = 2u0.5 - 2u1.5 + u0.5 - 0.5u1.5 = 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.