An equation can be simplified by doing the same thing to both sides: move a subtracted amount across so everything is positive (completion), then cancel matching amounts that appear on both sides (reduction). Keeping your eye fixed on the two sides at every step turns a tangled problem into a plain one.
Every equation made of squares, unknowns and plain numbers can be boiled down to one of six standard shapes, each with its own fixed recipe. Solving a new problem then means recognising which shape it reduces to, not inventing a fresh trick each time.
Once quantities are handled as unknowns obeying the same rules, whole numbers, awkward roots and geometric lengths can all be worked on in one way. This broke from the older view of mathematics as essentially geometry and let mathematics be applied to itself.