2023 · Q1
If a² − bc = α, b² − ac = β, c² − ab = γ, then what is (aα + bβ + cγ) / ((a + b + c)(α + β + γ)) equal to?
Elementary MathematicsAlgebra
- Aa + b − c
- Ba − b + c
- C−a + b + c
- D1
Explanation
Put alpha=a^2-bc, beta=b^2-ca, gamma=c^2-ab. Then a*alpha+b*beta+c*gamma = (a^3+b^3+c^3)-3abc = (a+b+c)(alpha+beta+gamma)/... ; in fact a^3+b^3+c^3-3abc = (a+b+c)((a^2+b^2+c^2)-(ab+bc+ca)) = (a+b+c)(alpha+beta+gamma). Dividing by (a+b+c)(alpha+beta+gamma) leaves 1. Option (d).