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When solving quadratic equations such as \((x+2)^2 - 4 = 0\), the first step is to make the equation easier to handle. Generally, the method involves making one side of the equation the square of a binomial. This particular equation is well set for using square roots.To begin, we want to isolate the quadratic term. In this exercise, you can do this by performing an algebraic operation. We can achieve this by adding or subtracting numbers to/from both sides. This balances the equation, maintaining equality. In our case, adding 4 to both sides gives us \((x+2)^2 = 4\). This new equation is prepared for the next step, which involves using square roots to solve for \(x\). The essence of solving such equations is to simplify them step by step until we can pinpoint the variable's value(s).

The square root step is crucial when solving an equation of this nature. When you have an equation like \((x+2)^2 = 4\), the objective is to get rid of the square by applying the square root.Taking the square root of both sides simplifies the left side by removing the square, yielding \(x+2\). Always remember to consider both the positive and negative roots of the number on the right side. Here, the square root of 4 can be either 2 or -2. This is because \((-2)\times(-2) = 4\) just as \(2\times2 = 4\).Thus, when you take the square root of both sides, remember to express this as:

• \(x+2 = 2\)
• \(x+2 = -2\)

These represent the potential solutions we need to consider in further steps.

Algebraic manipulation is a series of strategies including addition, subtraction, multiplication, and division to disentangle equations. After taking square roots, you acquired two linear equations: \(x+2=2\) and \(x+2=-2\).The goal is to isolate \(x\) in these equations. This usually involves simple subtraction or addition. For \(x+2=2\), subtract 2 from both sides to find \(x=0\). Similarly, for \(x+2=-2\), subtract 2 from both sides to get \(x=-4\).These steps are straightforward operations but require careful attention to ensure both conditions (positive and negative square roots) are explored. That's how you end up with solutions \(x=0\) and \(x=-4\), completing the exercise. Remember, methodology and precision are key in algebraic manipulation to solve equations correctly.