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Quadratic equations take the form of ax^2 + bx + c = 0. They are called "quadratic" because the highest power of the variable is squared. In many cases, these equations must be solved to find the value of the variable, which satisfies the equation. In our example, the equation is 25x^2 = 4, and our goal is to find the value of x. To solve quadratic equations, there are several methods, including factoring, completing the square, and using the quadratic formula. Additionally, in specific cases like this exercise, we can apply the square root method. This method is most effective when the quadratic equation can first be simplified into the form of x^2 = k. Understanding and choosing the proper method to solve these equations is crucial for a correct and efficient solution.

The square root method is a straightforward technique to solve quadratic equations of the form x^2 = k. This method assumes that once the equation is simplified to this form, we can directly take the square root of both sides to solve for x.
Here's how it works in steps:

• First, isolate the x^2 term on one side of the equation if it is not already done. In our exercise, we started with 25x^2 = 4, and then divided both sides by 25, resulting in x^2 = \(\frac{4}{25}\).
• Next, apply the square root to both sides. Remembering that taking a square root can give both a positive and a negative solution, we then find x = ±√(\(\frac{4}{25}\)).

This method is particularly efficient when the equation doesn't have a linear (bx) term. It's a neat and quick approach to find solutions when applicable.

Simplifying radical expressions is a crucial skill in algebra. It involves rewriting the expression in a cleaner and more understandable form. In solving the given equation, we reached the step where we had to simplify √(\(\frac{4}{25}\)).
Simplification can be broken down into:

• First, take the square root of the numerator and the denominator separately, giving us √4 and √25.
• Simplifying these, we know that √4 = 2 and √25 = 5.
• This results in the simplified form \(\frac{2}{5}\). So, the values of x become x = ± \(\frac{2}{5}\).

Simplifying radicals not only makes calculations more manageable but also helps in understanding and solving equations more quickly and accurately. It is a powerful tool that plays a significant role in solving quadratic equations efficiently.