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When faced with an equation involving a square root, the initial aim is to isolate the variable. This means we want to manipulate the equation so that we have the variable on one side and all other numbers on the other. The process of isolating the variable is crucial as it sets the stage for finding the solution.

For instance, in the equation \(\sqrt{3x-7}=0\), the variable \(x\) is contained within the square root. Our goal is to free \(x\) from the square root to solve for it. The first step is to square both sides to eliminate the square root. Once the square root is removed and we obtain a simpler equation \(3x-7=0\), we can focus on getting \(x\) by itself. We do this by performing inverse operations such as addition and division until \(x\) stands alone, as seen with \(x = 7/3\) in the given solution.

After isolating and solving for the variable, we must verify our solution by substituting it back into the original equation. This step is vital to ensure that the solution is not only mathematically correct but also makes sense within the parameters of the given problem.

In our example, we calculate by plugging \(x = 7/3\) back into the original equation: \(\sqrt{3(7/3) - 7} = 0\). Simplifying inside the square root should return us to the equation we started with if our solution is correct. We find that indeed, \(\sqrt{0} = 0\), validating our answer. Checking the solution not only confirms it but also helps identify any potential algebraic mistakes made along the way.

Squaring both sides of an equation is a common method used to eliminate a square root. By applying the same operation to both sides, we maintain the equilibrium of the equation while transforming it into a more manageable form.

The operation involves taking the exponent of 2, or squaring, applied to the entire side of the equation containing the square root. It is crucial to square the entire side to abide by the rules of operations within mathematics. As shown in the example \(\sqrt{3x-7} = 0\), we square the equation to get \(3x-7 = 0^2\), which simplifies to \(3x-7 = 0\). By doing so, we've successfully eliminated the square root and can proceed to isolate the variable. It's important to remember that whenever we square both sides, we must be cautious of potential extraneous solutions which can occur, especially when dealing with square roots and squared terms.