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?
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
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
note that log(y1)+log(y2)+log(y3) ≠ log(y1+y2+y3)
To find the sum of all real values of x that satisfy f(x) = 0, we need to solve the equation f(x) = 0. Let's break down the given function f(x) = 3^(171x−3)−3^(114x+2)+3^(57x+3)−1.
Step 1: Rewrite the equation
f(x) = 3^(171x−3)−3^(114x+2)+3^(57x+3)−1 = 0.
Step 2: Simplify the equation
Now, we can rearrange the terms and use the properties of exponents to simplify the equation:
3^(171x−3)−3^(114x+2)+3^(57x+3)−1 = 0
3^(171x−3) + 3^(57x+3) = 3^(114x+2) + 1
Step 3: Use the fact that a^b + a^c = a^b+c
By using the property of exponents, we can rewrite the equation as:
3^(171x−3) + 3^(57x+3) = 3^(114x+2) + 3^0
Since 3^0 = 1, we simplify further:
3^(171x−3) + 3^(57x+3) = 3^(114x+2) + 1
Step 4: Set the exponents equal to each other and solve
Now that the bases are the same, we can set the exponents equal to each other:
171x−3 = 114x + 2
Step 5: Solve for x
By isolating x in the equation, we find:
171x−114x = 2 + 3
57x = 5
x = 5/57
So, one of the real values of x that satisfies f(x) = 0 is x = 5/57.
Step 6: Find the remaining values of x
To find the remaining real values of x that satisfy f(x) = 0, we need to repeat steps 3-5 for the remaining terms in f(x) = 0.
By doing this, we can find out if there are any other real values of x that satisfy the equation.
After completing the steps, suppose we find two more real values of x, denoted as x1 and x2.
Step 7: Calculate the sum of all real values of x
Now, we can calculate the sum of all real values of x that satisfy f(x) = 0:
S = x + x1 + x2 + ...
Step 8: Calculate 1/S
To find the value of 1/S, we take the reciprocal of the sum S:
1/S = 1/(x + x1 + x2 + ...)
Therefore, the value of 1/S depends on the specific values of x, x1, x2, and so on, that satisfy f(x) = 0.