This theorem is equivalent to:

(a - b)² = a² - 2⋅a⋅b + b²

Here is a visualization of the theorem:

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

The area of the orange square is (a - b)².

Let's now try to describe the area of the orange square in terms of the other rectangles.

The area of the orange square is:

(The area of the largest square) - 2 ⋅ (the area of the blue rectangle) - (the area of the green square)

The area of the largest square is

a²

The area of the blue rectangle is

b ⋅ (a - b)

The area of the green rectangle is

b²

Then using substitution, the area of the orange square is

a² - 2 ⋅ (b ⋅ (a - b)) - b²

2 ⋅ (b ⋅ (a - b)) is equal to

(2 ⋅ b ⋅ a) - (2 ⋅ b ⋅ b)

Which is

(2 ⋅ b ⋅ a) - (2 ⋅ b²)

After substitution, we get:

a² - ((2 ⋅ b ⋅ a) - (2 ⋅ b²)) - b²

Which reduces to

a² - (2 ⋅ b ⋅ a) + b²

We can also prove this theorem using the Sum Squared Theorem, which used the distributive property, as shown below.

Quiz (1 point)

• (a + b) ⋅ (a + b) = ((a ⋅ a) + ((a ⋅ b) ⋅ 2)) + (b ⋅ b)
• (a ⋅ (b ⋅ (-1))) ⋅ 2 = (a ⋅ b) ⋅ (-2)
• (a ⋅ (-1)) ⋅ (a ⋅ (-1)) = a ⋅ a

• if a = b, then x + a = x + b

• if a = b, then a + c = b + c

• if a = b, then x + a = x + b

• if the following are true:

  • a = b
  • b = c

  then a = 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.