This theorem is equivalent to:

(a + b)² = a² + 2⋅a⋅b + b²

Here is a visualization of the theorem:

The area of the orange square is

a ⋅ a

The area of the blue rectangle is:

a ⋅ b

The area of the green square is

b ⋅ b

Then, the total area of 1 orange square, 2 blue rectangles, and 1 green square is:

(a ⋅ a) + 2 ⋅ (a ⋅ b) + (b ⋅ b)

The graph shows that we can arrange the 4 rectangles into a larger square. The side of this larger square is a + b. Therefore, the area of the larger square is:

(a + b) ⋅ (a + b)

The two areas are equal, so we claim that:

(a ⋅ a) + 2 ⋅ (a ⋅ b) + (b ⋅ b) = (a + b) ⋅ (a + b)

This theorem can also be proved fairly easily using the distributive property, as shown below.

Quiz (1 point)

• a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)
• (b + c) ⋅ a = (a ⋅ b) + (a ⋅ c)
• (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c)
• (a + b) + c = a + (b + c)
• a + a = a ⋅ 2

• if the following are true:

  • a = b
  • c = d

  then a + c = b + d

• if the following are true:

  • a = b
  • b = c

  then a = c

• if a + b = c, then ((x + a) + b) + y = (x + c) + y

• if the following are true:

  • a = b
  • a = c

  then b = c

• if the following are true:

  • a = b
  • b = c

  then a = c

Please write your proof in the table below. Each row should contain one claim. The last claim is the statement that you are trying to prove.

Step	Claim	Reason (optional)	Error Message (if any)
1
2
3
4
5
6
7
8
9
10

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