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One of the primary steps in solving radical equations, which involve square roots, is to isolate the square root on one side of the equation. Isolating the square root makes it easier to deal with because you can then perform operations that will eliminate the square root and leave you with a more familiar linear or quadratic equation.

In the example given, the equation \(\sqrt{2x - 3} - 3 = 0\) contains a square root that needs to be isolated. To do this, we perform the simple algebraic operation of addition to both sides, helping us move constants away from the radical. By adding 3 to both sides, the equation becomes \(\sqrt{2x - 3} = 3\). This crucial step sets up the stage for the next part of the process where we can eliminate the square root entirely.

After isolating the square root, the next essential step is to square both sides of the equation. Why do we square? Squaring the square root effectively removes it, as squaring is the inverse operation of square rooting. This allows us to transform a radical equation into a standard algebraic equation which is easier to solve.

In the equation from our example, \(\sqrt{2x - 3} = 3\), when we square both sides, the equation simplifies to \(2x - 3 = 9\). Squaring the left side eliminates the square root, and squaring the right side gives us a numerical value with which we can work more easily. Remember to square the entire expression on each side, not just the individual terms, to avoid common mistakes.

The final stretches of solving radical equations includes algebraic manipulation. This involves operations such as addition, subtraction, multiplication, division, or factoring, with the aim of finding the value of the variable.

Continuing from our squared equation \(2x - 3 = 9\), we proceed by adding 3 to both sides to isolate the term with the variable \(x\). This gives us \(2x = 12\). The last step is to divide both sides by 2, yielding \(x = 6\), the solution. It's paramount to perform each operation correctly and to both sides of the equation to maintain equality.

Algebraic manipulation may also require you to check for extraneous solutions, especially in equations involving radical terms, since squaring can sometimes introduce solutions that don't work in the original equation.