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Quadratic equations are mathematical expressions of the second degree, typically presented in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. To solve such equations, one can employ various methods, such as factoring, completing the square, using the quadratic formula, or the square root property, which is especially useful when the equation is in the form \(x^2 = k\).

When using the square root property for the given exercise, \(3x^2 - 1 = 47\), it is essential to isolate the term containing \(x^2\) on one side. Once \(x^2\) is isolated, taking the square root of both sides will yield two potential solutions since \(\sqrt{x^2}\) equals both \(x\) and \( - x\). It's crucial to include both the positive and negative roots to capture all possible solutions.

Rearranging equations is a foundational skill in algebra that allows us to isolate the variable we are solving for. To enhance clarity and minimize errors, it's recommended to perform one operation at a time and apply it to both sides of the equation. This ensures the equation remains balanced. Using the exercise \(3x^2 - 1 = 47\) as an example, the first rearrangement step is to add 1 to both sides to isolate the \(3x^2\) term. Then, dividing both sides by 3 yields \(x^2 = 16\), which sets the stage for applying the square root property.

In the context of the square root property, this step is crucial because it simplifies the equation into a form where the variable can be easily extracted by taking the square root. Careful arrangement also helps to avoid messy calculations and potential mistakes, leading to a cleaner and more straightforward solution process.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 since \(4 \times 4 = 16\). When solving quadratic equations by the square root property, it's essential to remember that every positive number actually has two square roots: a positive and a negative root.

In the example \(x^2 = 16\), applying the square root property allows us to deduce that \(x\) could either be \(4\) or \( - 4\) because both \(4 \times 4\) and \( - 4 \times - 4\) result in 16. Recognizing and including both square roots is a vital step towards solving quadratic equations completely. Eliminating either solution may discard a valid answer to the problem. Square root calculations like these are basic yet critical in unlocking quadratic equations, helping students to fully realize the set of potential solutions.