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.

Answer: x = 3 on PLATO