2016 · Q16
If the sum of the roots of ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then which one of the following relations is correct?
Elementary MathematicsAlgebra
- Aab² + bc² = 2a²c
- Bac² + bc² = 2b²a
- Cab² + bc² = a²c
- Da² + b² + c² = 1
Explanation
For roots α, β: α+β = −b/a and αβ = c/a. The sum of the squares of the reciprocals is 1/α²+1/β² = ((α+β)²−2αβ)/(αβ)² = (b²−2ac)/c². Equating to the sum of roots gives −b/a = (b²−2ac)/c², which rearranges to ab²+bc² = 2a²c, option A.