Problem Solving

What is the sum of all solutions to the equation 3x^(2/3) = 54 + 9x^(1/3)
A) 63
B) 189
C) 216
D) 243
E) 567

Answer: B

Source: GMAT Prep Now
Difficulty level: ≈700

Brent@GMATPrepNow wrote:What is the sum of all solutions to the equation 3x^(2/3) = 54 + 9x^(1/3)
A) 63
B) 189
C) 216
D) 243
E) 567

We should start by recognizing that this is a QUADRATIC EQUATION in disguise.

Notice that x^(2/3) = [x^(1/3)]²

So let's let x^(1/3) = k, and replace x^(1/3) with k to get: 3k² = 54 + 9k
Rearrange to get: 3k² - 9k - 54 = 0
Factor: 3(k² - 3k - 18) = 0
Factor more: 3(k - 6)(k + 3) = 0
So, the solutions are k = 6 and k = -3

Since k = x^(1/3), we can write: x^(1/3) = 6 and x^(1/3) = -3
If x^(1/3) = 6, then x = 6³ = 216
If x^(1/3) = -3, then x = (-3)³ = -27

So, the sum of all solutions = 216 + (-27) = 189

Answer: B

Cheers,
Brent