Problem
ALG-B1-M04-P024 Value at the Sum of Roots
#24
★★★★★ Level 5 of 5
The quadratic trinomial \(f(x)=ax^2+bx+c\) has two distinct real roots \(r_1,r_2\). Suppose \(f(r_1+r_2)=137\). Find \(c\).
Inspired by regional olympiad method · 2025 · Grade 10 · Problem 1
By Vieta, \(r_1+r_2=-\frac ba\). Substitute this value into \(f\).
By Vieta, \(r_1+r_2=-\frac ba\). Therefore
\[f(r_1+r_2)=f\left(-\frac ba\right)=a\frac{b^2}{a^2}-\frac{b^2}{a}+c=c.\]
Hence \(c=137\).
The strong move is to see the axis of symmetry or substitute the sum of roots.