This theorem is equivalent to:

a² - b² = (a + b) ⋅ (a - b)

Here is a visualization of the theorem:

In this graph, the side of the largest square is a. The side of the orange square is b. Thus, the longer side of the blue rectangle is a - b.

The difference between the area of the largest square and the area of the orange square is:

a² - b²

Visually, if we remove the orange square from the largest square, then we are left with 2 blue rectangles and the green square.

The area of 2 blue rectangles and the green square is:

2 ⋅ ((a - b) ⋅ b) + (a - b)²

If we factor out (a - b) from both terms, we get:

(2 ⋅ 1 ⋅ b + (a - b)) ⋅ (a - b)

(2 ⋅ 1 ⋅ b + (a - b)) is a + b. Therefore, the area of 2 blue rectangles and the green square is:

(a + b) ⋅ (a - b)

We can also prove this theorem using the distributive property, as shown below.

Quiz (1 point)

• a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)
• (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c)
• (a + b) ⋅ (b ⋅ (-1)) = ((a ⋅ b) ⋅ (-1)) + ((b ⋅ b) ⋅ (-1))
• (b ⋅ a) + ((a ⋅ b) ⋅ (-1)) = 0

• if the following are true:

  • a = b
  • c = d

  then a + c = b + d

• if b + c = 0, then (a + b) + (c + d) = a + d

• if the following are true:

  • a = b
  • b = c
  • c = d

  then a = d

• if a = b, then b = a

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)
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