Let the quote symbol (") indicate the squaring operation.
The first display in the grid to the right includes  the red perimeter of a square partitioned into four sub-areas by intersecting blue lines. The intersection separates the blue lines into two segments. The length of the longer segment is designated (a) and the length of the shorter segment is designated (b). Therefore the four sub-areas will be designated (a"), (b") and (ab) and the total area of the square bounded by red lines (a + b)" equals (a" +  b" + 2ab). This analysis of the area of a square gives us the quadratic formulation (a + b)" = (a" + 2ab + b"). Note areas designated 2ab comprise four yellow right triangles.

Click the NEXT AREA button. Now the four yellow right triangles are each nested in the four corners of the red lined square. Note that the areas of the yellow triangles do not overlap, and the longest sides of the triangles form the perimeter of an inner square. Since the areas of the four triangles equal 2ab, the area of the inner square must equal the total area of the red lined square minus 2ab or (a + b)" - 2ab. In expanded terms (a" + b" + 2ab -2ab) = (a" + b"). If we designate the lengths of the long sides of the right triangle (c) then we have (a" + b") = (c"). This formulation is the pythagorean theorem for right triangles.

Click the NEXT AREA button. Now the four yellow right triangles have been rotated into the (c") area in such a manner that there areas do not overlap. Note that the lengths of the sides of the inner most square equal the length (a) of the longest leg of the right triangles minus the length (b) of the shortest leg of the right triangles, (a - b). Since the areas of the four triangles equal 2ab, the area of the inner most square must equal the area of the middle inner square (a" + b") - 2ab. So we have the quadratic formulation (a - b)" = a" - 2ab + b".