To solve these equations, you generally need to isolate the square roots, square both sides to eliminate them, and solve for \(x\). Here’s the step-by-step solution for each:

### Equation 2:
\[
\sqrt{x-1} - \frac{1}{2}\sqrt{4x-4} + \sqrt{\frac{18x-18}{2}} = 6
\]

1. Simplify each square root.
2. Combine like terms.
3. Isolate one square root, square both sides, and solve for \(x\).

### Equation 4:
\[
6\sqrt{x-5} + \sqrt{9x-45} - 2\sqrt{4x-20} = 25
\]

1. Simplify each square root.
2. Try substituting \(y = \sqrt{x-5}\) to simplify terms.
3. Rearrange, solve for \(y\), and then for \(x\).

### Equation 6:
\[
\sqrt{9x+27} + 4\sqrt{x+3} - \frac{3}{4}\sqrt{16x+48} = 0
\]

1. Simplify each term under the radicals.
2. Isolate one square root and square the equation.
3. Repeat until you solve for \(x\).

### Equation 8:
\[
\sqrt{1-x} + \sqrt{4-4x} - \frac{1}{3}\sqrt{16-16x+5} = 0
\]

1. Simplify terms as much as possible.
2. Square each step to eliminate radicals systematically.
3. Solve for \(x\).

### Equation 10:
\[
\sqrt{9x-45} - \sqrt{x-5} = 4 - \frac{1}{2}\sqrt{4x-20}
\]

1. Simplify using substitutions.
2. Isolate square roots, square both sides, and simplify.
3. Solve for \(x\).

### Equation 12:
\[
\sqrt{x-3} + \sqrt{9x-27} - \frac{1}{2}\sqrt{4x-12} = 6
\]

1. Simplify each to standard form.
2. Use systematic removal of each square root.
3. Solve step-by-step for \(x\).

### Equation 14:
\[
\sqrt{16x+48} - 7\sqrt{x+3} + \frac{3}{4}\sqrt{4x+12} = -6
\]

1. Simplify each square root.
2. Square sides to eliminate radicals.
3. Keep terms balanced; solve for \(x\).

These problems involve trial and error, rearranging, or substitutions to simplify radicals, which vary case by case.