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Quadratic f with f(x + y) = f(x) + f(y) + c - kxy: find the constants and a sum of f values

The same question, asked 2 times: 29 Jun 2022*, 28 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    29 Jun 2022

    87
    Let c,k∈Rc,k\in R. If f(x)=(c+1)x2+(1−c2)x+2kf(x) = (c+ 1)x^{2}+\left( 1 -c^{2} \right)x+ 2k and f(x+y)=f(x)+f(y)−xyf(x+y) =f(x) +f(y) -xy, for all x,y∈Rx,y\in R, then the value of ∣2(f(1)+f(2)+f(3)+⋯+f(20))∣|2(f(1) +f(2) +f(3) + \cdots + f(20))| is equal to

  2. Question 2

    28 Jan 2025

    55
    Let f:R→Rf:R \rightarrow R be a function defined by f(x)=(2+3a)x2+(a+2a−1)x+b,a≠1f(x) = (2 + 3a)x^{2}+\left( \frac{a + 2}{a - 1} \right)x + b,a \neq 1. If f(x+y)=f(x)+f(y)+1−27xyf(x + y) = f(x) + f(y) + 1 -\frac{2}{7}xy, then the value of 28∑i=15∣f(i)∣28\sum_{i = 1}^{5} |f(i)| is:

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