Take a strip away

The idea

Replacing b by –b in (a + b)² = a² + 2ab + b² gives (a – b)² = a² – 2ab + b², also seen by removing two rectangles from a square of side a.

A square inside a square

A square has side a. Split a into two parts: a short part b and a long part a − b. The big square has area a². Inside it, the small square of side a − b sits in a corner. To get it, we cut rectangles off the big square.

How many rectangles must we cut away to leave the small square?

• 1
• 2
• 3

Cut the strips

Take away the long strip first, then the second strip. What is left is the small square. Do it for two different squares.

Two strips go

(a − b)² = a² − 2ab + b². Start with the big square a², take away the long strip ab and the strip b(a − b).

(a − b)² = a² − ab − b(a − b) = a² − ab − ba + b² = a² − 2ab + b² You get the same by writing b as −b in (a + b)² = a² + 2ab + b². Only the middle term changes sign, because (−b)² = +b².

Notes

(a − b)² = a² − 2ab + b². Cut the strips ab and b(a − b) off a square of side a, or write b as −b in (a + b)².

Check yourself

Work out a² − 2ab + b² for a = 10 and b = 3.

Answer: 49

100 − 60 + 9 = 49, which is also (10 − 3)² = 7².

Which is the expansion of (x − y)²?

Find 29² using (30 − 1)².

Answer: 841

900 − 60 + 1 = 841.

Expand (3x − 2)².

• x² − 2xy + y² — correct. Yes: x², take away two rectangles xy, then add back y².
• x² − y². The middle term −2xy is missing, and the last term is +y² (it is not subtracted).
• x² − 2xy − y². (−y)² = +y². The corner y × y was cut away twice, so it is added back once.
• 9x² − 4. The middle term −12x is missing.
• 9x² − 12x + 4 — correct. Yes: (3x)² = 9x², 2(3x)⁠(2) = 12x taken away, and (−2)² = +4.
• 9x² − 12x − 4. (−2)² = +4, so the last term is +4.