The sum of squares S = α^2 + β^2 + γ^2 + δ^2 can be written in terms of a, b, c as which of the following?

The sum of squares S = α^2 + β^2 + γ^2 + δ^2 can be written in terms of a, b, c as which of the following?

• A. (b^2 - 2ac)/a^2
• B. (b^2 + 2ac)/a^2
• C. (b^2 - 2ad)/a^2
• D. b^2/a^2 - 2c/a

Answer: (A)

Think about how the squares of the roots relate to the sums of the roots and the pairwise products. If α, β, γ, δ are roots of a polynomial with leading coefficient a, then the sum of the roots is -b/a and the sum of all pairwise products is c/a. The sum of squares is S = (α+β+γ+δ)^2 - 2(αβ+αγ+αδ+βγ+βδ+γδ) = (-b/a)^2 - 2(c/a) = b^2/a^2 - 2c/a = (b^2 - 2ac)/a^2. This matches the given form.

Think about how the squares of the roots relate to the sums of the roots and the pairwise products. If α, β, γ, δ are roots of a polynomial with leading coefficient a, then the sum of the roots is -b/a and the sum of all pairwise products is c/a. The sum of squares is

S = (α+β+γ+δ)^2 - 2(αβ+αγ+αδ+βγ+βδ+γδ)

= (-b/a)^2 - 2(c/a)

= b^2/a^2 - 2c/a

= (b^2 - 2ac)/a^2.

This matches the given form.