A square cut in four
Identity for (a + b)²
Cut the square
A square has sides made of two pieces: a piece of length a and a piece of length b. So every side has length a + b. Draw one line across the square and one line down, at the joins.
Into how many pieces does the square fall?
What this lesson covers
The idea
(a + b)² = a² + 2ab + b²: a square of side a + b splits into squares of area a² and b² and two rectangles of area ab, and the distributive property shows it holds for all numbers.
Cut the square
A square has sides made of two pieces: a piece of length a and a piece of length b. So every side has length a + b. Draw one line across the square and one line down, at the joins.
Into how many pieces does the square fall?
- 2
- 3
- 4
Cut it up
Change a and b. The big square falls into four pieces. Does the whole area match the pieces added up?
Four pieces
(a + b)² = a² + 2ab + b²: the square of side a + b is a square a², a square b², and two rectangles ab.
Why does it hold for every number, not only for lengths? The distributive property shows it: (a + b)(a + b) = a(a + b) + b(a + b) = a² + ab + ba + b² = a² + 2ab + b²
Notes
(a + b)² = a² + 2ab + b² for all numbers a and b. A square of side a + b splits into squares a² and b² and two rectangles ab.
Check yourself
Take a = 3 and b = 2. The two rectangles together have area 2ab. What is 2ab?
Answer: 12
One rectangle has area 3 × 2 = 6, and there are two of them: 2ab = 12.
Which is the expansion of (x + y)²?
Does it work for negative numbers? Take a = −2 and b = −3. What is a² + 2ab + b²?
Answer: 25
a² = 4, b² = 9 and 2ab = +12, because a negative times a negative is positive. The total is 4 + 12 + 9 = 25, and (a + b)² = (−5)² = 25 too.
When does (a + b)² = a² + 2ab + b² hold?
- x² + y². This leaves out the two rectangles of area xy.
- x² + xy + y². There are two rectangles of area xy, so the middle term is 2xy.
- x² + 2xy + y² — correct. Yes: the square x², two rectangles xy, and the square y².
- For all numbers a and b — correct. Yes. The picture shows it for lengths and the distributive property shows it for every number.
- Only when a and b are lengths. The distributive proof also works for negative numbers and fractions.
- Only when a and b are whole numbers. The distributive property does not care what kind of numbers a and b are.