A square cut into four pieces
A square yard has a side of a tiles. Its side grows by b tiles, to a + b. Its new area is the side times itself: (a + b)². That area is not a² + b².
a + b, whole. Step 1 of 4
- Square of side
a + b, whole. - Cut it: top-left corner is
a². - Two matching rectangles, each
ab. - Bottom-right corner is
b². Total =a² + 2ab + b².
Rule: (a + b)² = a² + 2ab + b². The two ab rectangles are real pieces, not extras.
Take a = 5 and b = 2. The side is 5 + 2 = 7. So the area is 7 × 7 = 49. The pieces give 25 + 10 + 10 + 4, also 49.
A big number works too. Cut 85 as 80 + 5. Then 85² = 80² + 2 × 80 × 5 + 5². That is 6,400 + 800 + 25 = 7,225.
For a difference, start with the square a². Take away two strips, each a by b. Their shared b by b corner is taken away twice. So add b² back: (a − b)² = a² − 2ab + b². At a = 5 and b = 2, that is 25 − 20 + 4 = 9. And 3 × 3 = 9.
Pick a way to split 85. Count the four blocks.
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