Problem Solving

Vincen wrote:If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81

Given: 3^x - 3^(x-1) = 162
Factor to get: [3^(x-1)][3^1 - 1] = 162
Simplify to get: [3^(x-1)][2] = 162
Divide both sides by 2 to get: 3^(x-1) = 81
Rewrite the right side as 3^(x-1) = 3^4
So, x - 1 = 4
This means x = 5

We get x(x - 1) = (5)(5 - 1) = (5)(4) = 20

Answer: C

Cheers,
Brent

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