What are the solutions for x in the equation x - 3 = 6sqrt((2/3)x - 2)?

To solve the equation ( x - 3 = 6\sqrt{\left(\frac{2}{3}x - 2\right)} ), we begin by isolating the square root term and then squaring both sides to eliminate the square root.

Here is the step-by-step process:

1. Start with the equation:

[

x - 3 = 6\sqrt{\left(\frac{2}{3}x - 2\right)}

]

2. Isolate the square root:

[

\sqrt{\left(\frac{2}{3}x - 2\right)} = \frac{x - 3}{6}

]

3. Square both sides:

[

\frac{2}{3}x - 2 = \left(\frac{x - 3}{6}\right)^2

]

4. Expanding the right-hand side, we get:

[

(x - 3)^2 = x^2 - 6x + 9, \quad \text{and thus } \left(\frac{x - 3}{6}\right)^2 = \frac{x^2 - 6