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When solving quadratic equations, a fundamental step is to isolate the variable. To make the variable the subject of the formula, you want it to be on one side of the equation all by itself. In our example, the equation starts as \(2x^2 = 250\). To focus on the variable \(x\), we begin by dividing both sides by 2, yielding \(x^2 = 125\). It's essential to perform the same operation on both sides of the equation to maintain balance, which is the core of equation-solving.

Remember, quadratic equations typically end up with the variable squared, so isolating the variable often means getting to a point where you have \(x^2\) alone on one side. From there, you can further simplify to solve for \(x\). Isolating the variable is a critical strategy not just for quadratic equations, but for all algebraic equations, as it lays the groundwork for the solution.

Once the variable is isolated and we have \(x^2 = 125\), the next step in solving a quadratic equation is often taking square roots. This step allows us to deal with the square of the variable and to simplify the equation to solve for \(x\) itself. However, an important concept when taking the square root of both sides is to remember that a positive number has two square roots: one positive and one negative.

So, when you apply the square root to both sides of our equation, it becomes \(x = \sqrt{125}\) and \(x = -\sqrt{125}\). It's crucial to include both roots because both are valid solutions to our original quadratic equation. Ignoring one of them would miss a potential solution. Understanding how to properly take square roots is integral to solving quadratic equations, and reinforces the concept that each step must maintain the equation's balance and integrity.

Finally, we get to the heart of solving quadratic equations — finding the quadratic equation solutions. A quadratic equation typically results in two solutions because when graphed, it creates a parabola which can intersect the x-axis at two points. After isolating the variable and taking the square root, as we did with our example yielding \(x = \sqrt{125}\) and \(x = -\sqrt{125}\), the final step involves calculating these roots to find the exact solutions.

In our case, after finding the square roots, the solutions are approximately \(x = 11.18\) and \(x = -11.18\) when rounded to two decimal places. It's vital to remember that the solutions to quadratic equations can be real or complex numbers and that the quadratic formula can also be used to find these solutions when the equation does not factor easily. The process illustrates the beauty of algebra - where abstraction through symbols leads to concrete answers, and each step in the process serves a purpose in unveiling the final solution.