I believe that the answer to the square root of x^2 + 3x = x-3 is X = 3, x=-3 but I'm not sure. I believe the answer to the square root of 8x -2 = the square root of 2x is x=2 again I'm not sure.

why are you posting under several names?

I like making up names and have always been told never to give my real name over the internet. Sorry it won't happen again.

To solve the equation sqrt(x^2 + 3x) = x - 3, we can follow these steps:

Step 1: Isolate the square root on one side.
Add 3 to both sides: sqrt(x^2 + 3x) + 3 = x

Step 2: Square both sides to eliminate the square root.
Square both sides of the equation: [sqrt(x^2 + 3x) + 3]^2 = x^2

Step 3: Simplify the equation.
When we expand the squared expression on the left side, we get:
x^2 + 3x + 9 + 6sqrt(x^2 + 3x) = x^2

Notice that the x^2 term cancels out, and we're left with:
3x + 9 + 6sqrt(x^2 + 3x) = 0

Step 4: Isolate the square root.
Subtract 3x and 9 from both sides: 6sqrt(x^2 + 3x) = -3x - 9

Step 5: Divide both sides by 6 to isolate the square root.
(6 / 6) * sqrt(x^2 + 3x) = (-3x - 9) / 6
sqrt(x^2 + 3x) = (-1/2)x - (3/2)

Step 6: Square both sides again.
Square both sides of the equation: [sqrt(x^2 + 3x)]^2 = [(-1/2)x - (3/2)]^2
x^2 + 3x = (1/4)x^2 + 3x + (9/4)

Step 7: Simplify the equation.
Multiply every term by 4 to eliminate the fractions:
4x^2 + 12x = x^2 + 12x + 9

Subtract (x^2 + 12x) from both sides:
3x^2 = 9

Step 8: Divide both sides by 3 to isolate x^2.
(3x^2) / 3 = 9 / 3
x^2 = 3

Step 9: Take the square root of both sides.
We have two possible solutions: x = sqrt(3) and x = -sqrt(3).

Therefore, the correct answers to the equation sqrt(x^2 + 3x) = x - 3 are x = sqrt(3) and x = -sqrt(3).

Regarding your second equation, sqrt(8x - 2) = sqrt(2x), to solve it, we follow the same steps:

Step 1: Isolate the square roots on one side:
Subtract sqrt(2x) from both sides: sqrt(8x - 2) - sqrt(2x) = 0

Step 2: Square both sides to eliminate the square roots:
Square both sides of the equation: [sqrt(8x - 2) - sqrt(2x)]^2 = 0^2
8x - 2 - 2sqrt(8x - 2)sqrt(2x) + 2x = 0

Step 3: Simplify the equation:
Combining like terms, we have:
10x - 2 - 2sqrt(16x^2 - 8x) = 0

Step 4: Isolate the square root:
Add 2 to both sides: 10x - 2 + 2 = 2sqrt(16x^2 - 8x)
10x = 2sqrt(16x^2 - 8x) + 2

Step 5: Divide both sides by 2 to isolate the square root:
(10x) / 2 = (2sqrt(16x^2 - 8x) + 2) / 2
5x = sqrt(16x^2 - 8x) + 1

Step 6: Square both sides again:
Square both sides of the equation: (5x)^2 = [sqrt(16x^2 - 8x) + 1]^2
25x^2 = 16x^2 - 8x + 2sqrt(16x^2 - 8x) + 1

Step 7: Simplify the equation:
Collect like terms: 9x^2 + 8x - 1 = 2sqrt(16x^2 - 8x)

Step 8: Square both sides yet again:
Square both sides of the equation: [9x^2 + 8x - 1]^2 = [2sqrt(16x^2 - 8x)]^2

Expanding and simplifying, we get:
81x^4 + 144x^3 - 2x^2 + 64x^2 - 16x + 8x - 1 = 4(16x^2 - 8x)

Further simplifying, we have:
81x^4 + 144x^3 + 62x^2 - 8x - 1 = 64x^2 - 32x

Step 9: Rearrange the terms:
Move all the terms to the left side of the equation: 81x^4 + 144x^3 - 62x^2 + 24x - 1 = 0

At this point, the equation becomes more complicated, and it may not have a simple solution. You could use numerical methods or software to find an approximate solution.