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When solving equations, especially those involving square roots or other nonlinear operations, we may encounter extraneous solutions. These are solutions that arise from the process of solving the equation but do not satisfy the original equation once substituted back. This can happen when both sides of an equation are squared, as it may introduce false solutions.

To avoid mistakes, always substitute your solutions back into the original equation to verify their validity. In our exercise, after finding the solution for the equation \(\sqrt{x} - 15 = 0\), we plug \(x = 225\) back into the original formula. Since it satisfies the condition \(\sqrt{225} - 15 = 0\), \(x = 225\) is not extraneous and is indeed a valid solution.

Remember to always verify solutions, particularly for equations that have involved altering their original form.

The square root function, represented as \(\sqrt{x}\), is a mathematical operation that returns a value which, when multiplied by itself, gives the original number \(x\). It is only defined for non-negative values of \(x\) in the real number system, as negative numbers do not have real square roots.

In our given problem, the function \(\sqrt{x}\) appears as part of the equation \(\sqrt{x} - 15 = 0\). Solving this involves finding a value of \(x\) where the expression evaluates correctly. Since the square root function is being isolated, it suggests looking for a non-negative real number solution. This is resolved by squaring both sides to simplify and remove the root operation, leading to a solvable linear form.

Isolating the variable is a crucial step in solving equations. It involves rearranging the equation to get the variable of interest, in our case \(x\), by itself on one side of the equation. This is often the first step because it simplifies what you are working with and makes further manipulations easier.

In the example \(\sqrt{x} - 15 = 0\), our first task is to isolate \(\sqrt{x}\) by making sure it stands alone on one side. Since it already does so in this equation, the next logical step is to eliminate the square root to solve for \(x\).

This is done by squaring both sides, resulting in \(x = 225\). If the equation were more complex, it might involve additional algebraic steps like adding, subtracting, multiplying, or dividing terms. Isolating the variable simplifies the complexity and helps you focus on finding the right answer.