Vincen wrote:If 3^x - 3^(x-1) = 162, then x(x - 1) =
A. 12
B. 16
C. 20
D. 30
E. 81
The OA is C.
We can simplify the left side of the equation by factoring out a common 3^x from both terms, and then factor 162 as 3^4 * 2^1. Then we have:
3^x - 3^x * 3^-1 = 3^4 * 2^1
3^x(1 - 3^-1) = 3^4 * 2^1
On the left side, note that the expression 1 - 3^-1 = 1 - 1/3 = 2/3. We now have:
3^x(2/3) = 3^4 * 2^1
3^x = (3^4 * 2)(3/2)
3^x = 3^4 *3
3^x = 3^5
x = 5
So, x(x-1) = 5(4) = 20.
Alternate Solution:
Note that 3^x = 3 * 3^(x - 1). Then the left hand side of the equation becomes:
3^x - 3^(x - 1) = 3 * 3^(x - 1) - 3^(x - 1)
Let's factor the common 3^(x - 1):
3 * 3^(x - 1) - 3^(x - 1) = 162
3^(x - 1)(3 - 1) = 162
3^(x - 1)(2) = 162
3^(x - 1) = 81
3^(x - 1) = 3^4
x - 1 = 4
x = 5
Then, x(x - 1) = 20.
Answer: C
