What is the solution to the equation below? 3/x-2+6=square root x-2 +8

Solution of the equation is

x = {1, (1 + √(-3))/2, (1 - √(-3))/2}

Step-by-step explanation:

Given equation:

3/(x-2) + 6 = √(x-2) + 8

Subtracting 8 from both sides:

3/(x-2) + 6 - 8 = √(x-2) + 8 - 8

3/(x-2) - 2 = √(x-2)

Taking square on both sides:

9/(x-2)² - 4 = x - 2

Adding 2 on both sides:

9/(x-2)² - 2 = x

Multiplying by (x-2)² on both sides:

9 - 2(x-2)² = x(x-2)²

9 - 2(x²- 2x + 4) = x(x²-2x+4)

9 - 2x² + 4x - 8 = x³ - 2x² + 4x

Adding 2x² - 4x on both sides:

9 - 8 = x³

x³ = 1

Taking cube root:

x = ∛1

The solution of the equation is thus the three cube roots of unity:

1, (1 + √(-3))/2 and (1 - √(-3))/2.