91Ó°ÊÓ

Understanding how to isolate the square root is essential for solving radical equations. Consider the equation \( \sqrt{x} - 9 = 0 \). The first step is to get the square root term, \( \sqrt{x} \), by itself on one side of the equation so that we can deal with it directly. This is done by performing inverse operations that cancel out any other terms. In our example, we do this by adding 9 to both sides. The equation then becomes \(\sqrt{x} = 9\), neatly isolating the square root on one side.

Isolation of the square root allows us to more smoothly apply the necessary operations to solve for the variable inside the square root. Without this step, attempts to solve the equation could become muddled or we might inadvertently violate the properties of square roots and operations on inequalities.

Once the square root has been isolated, squaring both sides of the equation becomes the logical next step. Squaring a number and taking the square root are opposite operations; when applied consecutively, they cancel each other out. This leaves the variable of interest, \(x\), by itself. In our exercise, we square the isolated equation, resulting in \(\left(\sqrt{x}\right)^2 = 9^2\).

The equation simplifies to \(x=81\). Squaring both sides is a critical procedure that eliminates the square root and provides an equation that can be easily solved. Remember that when squaring a square root, you directly obtain the argument of the square root, since \(\left(\sqrt{x}\right)^2 = x\). However, caution is necessary because squaring is not a 'symmetric' operation and can introduce extraneous solutions that must be checked, which brings us to the final concept.

Verifying the solution is a crucial, but often overlooked step in solving algebraic equations. Once you've solved for the variable, like we did with \(x=81\), you must ensure that it satisfies the original equation. Some operations, such as squaring both sides, can introduce additional solutions, also known as extraneous solutions, that are not valid for the initial equation. In the case of our problem, we confirm our solution by substituting \(x\) back into the original equation: \(\sqrt{81} - 9 = 0\), simplifying to \(9 - 9 = 0\), which is true.

Thus, \(x=81\) is a valid solution. Without this verification step, it's possible to claim incorrect solutions which do not solve the original problem. It is a good habit to check every potential solution by plugging it back into the original problem to ensure its validity.