It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.
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It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.