Question

What is the solution to the equation squareroot n= squareroot 3n-3 +1
n = –4, n = –1
n = –1
n = 1
n = 1, n = 4

Answer 1

Solve sqrt(n) = sqrt(3n−3) + 1.

Domain: need n ≥ 0 and 3n−3 ≥ 0 ⇒ n ≥ 1.

Isolate and square: sqrt(n) − 1 = sqrt(3n−3). Squaring,
n − 2√n + 1 = 3n − 3 ⇒ −2n − 2√n + 4 = 0 ⇒ n + √n − 2 = 0.
Let t = √n ≥ 0: t^2 + t − 2 = 0 ⇒ (t+2)(t−1) = 0 ⇒ t = 1 (t = −2 rejected).
So √n = 1 ⇒ n = 1. Check: √1 = √0 + 1 ⇒ 1 = 0 + 1, true.

Answer: n = 1.