Question

Let f(x)=3^(171x−3)−3^(114x+2)+3^(57x+3)−1. Let S be the sum of all real values of x that satisfy f(x)=0.
What is the value of 1/S?

Answer 1

3^(57x) = y

27 y - 9 y^2 + 1/27 y^3 - 1 = 0

y^3 - 3^5 y^2 + 3^6 y - 3^3 = 0

(y-y1)(y-y2)(y-y3) = 0

-y1 y2 y3 = -3^3

Log(y1) + Log(y2) + Log(y3) = 3 Log(3)

57 Log(3) S = 3 Log(3)

S = 1/19

Answer 2

The sum of the roots of ax^3+by^2+cy+d=0 is -b/a = 243

Note that the roots are
3 and 120±√1599

Answer 3

note that log(y1)+log(y2)+log(y3) ≠ log(y1+y2+y3)