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Isolating variables is a critical first step in solving algebraic equations, particularly quadratic equations. It involves rearranging the equation so that one side contains only variables and the other side contains only numbers. This creates a clearer path to finding the solution.

For example, to solve the quadratic equation in the given exercise, one must first isolate the squared term, which is achieved by dividing both sides by the constant that multiplies the variable squate, in this case, 2. After performing the operation, we get a much simpler equation: \(x^2 = 125\). This technique simplifies complex equations and lays the groundwork for further operations such as taking the square root, which is the next step towards finding the solution.

Understanding this step is essential as it often simplifies the equation, making the subsequent steps easier. Remember, for every operation you perform, it must be done to both sides of the equation to maintain its balance.

The square root method is a powerful way to solve quadratic equations after one has successfully isolated the variable term. When a quadratic equation is formulated as \(x^2 = c\), where \(c\) is a constant, taking the square root of both sides helps find the value of \(x\).

However, an important aspect of this method is to remember that every positive number has two square roots: one positive and one negative. This is why, in the given exercise solution, we find two potential values for \(x\): \(x = \sqrt{125}\) and \(x = -\sqrt{125}\). The reason we take both the positive and negative roots is that squaring either the negative or the positive version of the value will result in the positive constant we started with. Even though \(\sqrt{125}\) is not a perfect square, simplifying it is not always necessary for finding a viable solution.

Algebraic solutions encompass various methods used to solve algebraic equations with one or more variables. For quadratic equations, in particular, there are multiple techniques to find the solution, such as factoring, completing the square, or using the quadratic formula.

The choice of method often depends on the structure of the equation and the preference of the person solving it. In the step by step solution for the exercise, we used the square root method due to the equation's ready-to-square-root form after isolating \(x^2\).

It's important to recognize that whichever algebraic solution method is chosen, a fundamental understanding of manipulating equations and performing arithmetic operations will be crucial. Thus, cementing these basic skills will make approaching quadratic equations, and algebra as a whole, far more manageable.