A strip cut from a square
A verandah is a square, 7 tiles on each side. A strip 2 tiles wide is cut from two edges. How many tiles are left? Slide the strip.
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- A square of side a = 7, with a strip of width b = 2.
- Leftover corner: side a − b, area 25.
- Top strip: (a − b) × b = 10.
- Side strip: (a − b) × b = 10.
- Small corner: b × b = 4.
- All four pieces add to a² = 49.
- So (a − b)² = a² − 2ab + b².
The rule
25 tiles are left: a square with side 7 − 2 = 5.
(a − b)² = a² − 2ab + b². Here a is the big square's side. The letter b is the strip's width. In words: start with a², take away 2ab, then add b².
Taking away two full strips removes the small corner twice. So the corner is added back once.
At a = 7 and b = 2 the leftover is 5² = 25. The rule agrees: 49 − 28 + 4 = 25.
Using it
Find 38². Write 38 as 40 − 2. Take a = 40 and b = 2. Then 38² = 40² − 2 × 40 × 2 + 2². That is 1600 − 160 + 4 = 1444.
Reading it backwards
Is x² − 10x + 25 a square, meaning a bracket times itself? Check the two ends and the middle. The ends are the first and last terms.
| Expression | Both ends squares? | Middle = twice their product? | Verdict |
|---|---|---|---|
x² − 10x + 25 | yes, x and 5 | 2 × 5 = 10, and it reads −10 | (x − 5)² |
x² + 6x + 9 | yes, x and 3 | 2 × 3 = 6, and it reads +6 | (x + 3)² |
x² − 9x + 25 | yes, x and 5 | 2 × 5 = 10, but it reads −9 | not a square |
x² − 10x + 24 | no, 24 is not a square | nothing to check | not a square |
Watch out!
Writing (a − b)² as a² − b² takes away only the small corner. It never takes away the strips. At a = 7 and b = 2 it gives 45, not 25.