α^2 + β^2 equals which of the following?

α^2 + β^2 equals which of the following?

• A. (α+β)^2 - 2αβ
• B. (α+β)^2 + 2αβ
• C. α^2 + β^2 + 2αβ
• D. α^2 - β^2

Answer: (A)

When you expand the square of a sum, you get (α+β)^2 = α^2 + 2αβ + β^2. If you subtract 2αβ from both sides, you obtain α^2 + β^2 = (α+β)^2 − 2αβ. So the expression α^2 + β^2 is exactly the square of the sum minus twice the product. The other forms either add the cross term back in or give a different expression altogether (like α^2 − β^2), which isn’t generally equal to α^2 + β^2. Hence the correct representation is (α+β)^2 − 2αβ.

When you expand the square of a sum, you get (α+β)^2 = α^2 + 2αβ + β^2. If you subtract 2αβ from both sides, you obtain α^2 + β^2 = (α+β)^2 − 2αβ. So the expression α^2 + β^2 is exactly the square of the sum minus twice the product. The other forms either add the cross term back in or give a different expression altogether (like α^2 − β^2), which isn’t generally equal to α^2 + β^2. Hence the correct representation is (α+β)^2 − 2αβ.