Which expression equals α^2 + β^2 + γ^2 in terms of (α+β+γ)^2 and pairwise sums?

Which expression equals α^2 + β^2 + γ^2 in terms of (α+β+γ)^2 and pairwise sums?

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

Answer: (B)

The key idea is the expansion of the square of a sum: (α+β+γ)^2 = α^2 + β^2 + γ^2 + 2(αβ + βγ + αγ). To get α^2 + β^2 + γ^2 by itself, subtract the cross terms 2(αβ + βγ + αγ) from both sides. This yields α^2 + β^2 + γ^2 = (α+β+γ)^2 − 2(αβ + βγ + αγ). This form writes the sum of the squares in terms of the square of the total and the sum of the pairwise products, which is exactly what’s asked. Quick check: if α=β=γ=1, the left is 3 and the right is 9 − 2·3 = 3.

The key idea is the expansion of the square of a sum: (α+β+γ)^2 = α^2 + β^2 + γ^2 + 2(αβ + βγ + αγ). To get α^2 + β^2 + γ^2 by itself, subtract the cross terms 2(αβ + βγ + αγ) from both sides. This yields α^2 + β^2 + γ^2 = (α+β+γ)^2 − 2(αβ + βγ + αγ). This form writes the sum of the squares in terms of the square of the total and the sum of the pairwise products, which is exactly what’s asked. Quick check: if α=β=γ=1, the left is 3 and the right is 9 − 2·3 = 3.