### Instruction: Find $x,$ given that \[\sqrt{x+7} - \sqrt{x} = \sqrt{3}.\].### Response: I notice that this equation involves square roots on both sides, so I want to isolate one of them and then square both sides to get rid of it.
I'll start by adding $\sqrt{x}$ to both sides, which gives me \[\sqrt{x+7} = \sqrt{x} + \sqrt{3}.\]
Now I'll square both sides, which means I have to use the formula $(a+b)^2 = a^2 + 2ab + b^2$ for the right side. This gives me \[x+7 = x + 2\sqrt{3x} + 3.\]
Next, I want to simplify this equation by moving everything to one side and combining like terms. I'll subtract $x$ and $3$ from both sides, which gives me \[4 = 2\sqrt{3x}.\]
To get rid of the square root on the right side, I'll square both sides again. This gives me \[16 = 12x.\]
Finally, I'll divide both sides by $12$ to get \[x = \frac{4}{3}.\]