Two brackets, and every pairing between them
A square garden has sides of x metres. One side grows by 2 m. The other side grows by 3 m. How does its area grow?
- (x + 2)(x + 3): a square plus two strips.
- Old square: x².
- Long strip: 2x.
- Wide strip: 3x.
- Corner: 6.
- Add them: x² + 5x + 6.
Two brackets, and every pairing between them
The new area is x² + 5x + 6 square metres. The area grew by 5x + 6 square metres.
Rule: every term in one bracket meets every term in the other. Two terms against two terms give four products, all added.
Check it at a value. At x = 4, (4 + 2)(4 + 3) = 6 × 7 = 42. And x² + 5x + 6 = 16 + 20 + 6 = 42.
Numbers work the same way. Take a block of 6 rows by 8 columns. Add 2 rows and 3 columns. Then (6 + 2)(8 + 3) = 8 × 11 = 88. The parts are 6 × 8 = 48 and 6 × 3 = 18. The rest are 2 × 8 = 16 and 2 × 3 = 6. Their sum is 88.
Watch out!
Reading (x + 2)(x + 3) as x² + 6 misses the two middle strips. The mistake pairs first with first: x by x. It pairs last with last: 2 by 3. At x = 4 that reading gives 22. The real answer is 42.
Slide m (rows added) and n (columns added). Count the four parts.
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