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Extracting square roots is a fundamental technique in solving quadratic equations where the variable is squared (like in the equation \( x^{2} = 169 \)). The process begins with transforming the equation into the form \( x^{2} = a \) where \( a \) is a non-negative number. The next step involves applying the square root to both sides which, conceptually, is like asking the question: 'What number, when multiplied by itself, gives you \( a \)?' For the equation at hand, taking the square root of both sides leads to \( x = \pm\sqrt{169} \).

It's crucial to understand that this operation effectively reverses squaring the variable \( x \) and hence, we consider both positive and negative square roots because squaring either a positive or negative number yields a positive result. This is why the symbol \( \pm \) is used, representing that there are two potential square roots: one positive and one negative.

Once the square roots have been extracted, the next phase is square root simplification. This involves identifying the number which, when squared, would yield the value under the square root. For the given example \( \sqrt{169} \), you must determine which number multiplied by itself gives 169. Recognizing that \( 13 \times 13 = 169 \), it can be deduced that \( \sqrt{169} = 13 \).

Performing this simplification is essential as it turns the abstract notion of \( \sqrt{a} \) into a definite number, making it easier to handle in further calculations. Simplifying a square root requires knowing the perfect squares or having a method to break down the number into its prime factors, if possible. The primary goal is to express the root in its simplest form, which, for whole numbers, is the largest number that can be squared to yield the value within the radical.

In solving equations by extracting square roots, acknowledging the existence of both positive and negative roots is crucial. A common misconception is to only take the positive root, but because the original equation includes a squared variable, it means that either a positive or negative number could work as a solution.

For the equation \( x^{2} = 169 \), when we apply the square root to both sides, we get \( x = \pm 13 \). The \( +13 \) represents the positive root, while the \( -13 \) represents the negative root. It's easy to overlook the negative solution, but including both roots assures that we capture all possible solutions to the quadratic equation.

Why Include Both Roots?

Including both the positive and negative roots acknowledges the fundamental nature of squaring a number: that both the negation and the original number have the same square. For instance, both \( (13)^2 \) and \( (-13)^2 \) yield 169, and so, both numbers must be recognized as solutions to the equation \( x^{2} = 169 \) to fully solve it.