how do you solve 8-3x=the square root of 4xsquared+ the square root of 20 + 8

To solve the equation 8 - 3x = √(4x^2 + √20 + 8), we need to isolate the variable x.

Let's break it down step by step:

1. Start by moving the constant term (8) to the other side of the equation by subtracting 8 from both sides:
8 - 3x - 8 = √(4x^2 + √20 + 8) - 8
-3x = √(4x^2 + √20) - 8

2. Now, let's square both sides of the equation to eliminate the square root on the right side:
(-3x)^2 = (√(4x^2 + √20) - 8)^2
9x^2 = 4x^2 + √20 - 16√(4x^2 + √20) + 64

3. Combine like terms by moving all the terms with x squared to one side and all the constant terms to the other side:
9x^2 - 4x^2 = √20 - 16√(4x^2 + √20) + 64
5x^2 = √20 - 16√(4x^2 + √20) + 64

4. Move all terms with square roots to one side of the equation:
5x^2 + 16√(4x^2 + √20) = √20 + 64

5. Isolate the term with the square root by subtracting √20 from both sides:
5x^2 + 16√(4x^2 + √20) - √20 = 64

6. At this point, you can square both sides of the equation to eliminate the remaining square root on the left side:
(5x^2 + 16√(4x^2 + √20) - √20)^2 = 64^2

7. Simplify both sides of the equation. It may involve expanding the square and simplifying square roots:
25x^4 + 80x^2(4x^2 + √20) - 20(5x^2) + 2(16√(4x^2 + √20)(-√20)) = 4096

8. Continue simplifying and solving for x. Since these steps may involve quite complex calculations, it would be more convenient to use computational software or a graphing calculator to find the solution.

Remember to double-check your solution once you find it, as the process involves multiple steps and calculations that may introduce errors.