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A quadratic equation is a second-degree polynomial equation in a single variable x, with a non-zero coefficient for the term with the highest degree. It has the standard form: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The term \(ax^2\) represents the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.

In the example you provided, the equation \(x^2 = 9\) is a simplified quadratic equation where the linear term and the constant term are both zero. This equation is actually even briefer than the standard form and can be seen as \(x^2 - 9 = 0\) by bringing the constant term to the left side of the equal sign. Solving quadratic equations can be done through multiple methods, such as factoring, completing the square, using the quadratic formula, or, as in this case, by taking the square root of both sides.

A square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a number \(x\) is denoted by \(\sqrt{x}\). For example, the square root of 9 is 3, because \(3^2 = 3 \times 3 = 9\).

When you apply the square root function to both sides of a quadratic equation like \(x^2 = 9\), you are essentially asking, 'What number squared will give me 9?'. It is crucial to remember that every positive number actually has two square roots: a positive and a negative root. This is because both positive and negative numbers when squared will give a positive result. For instance, \(3^2 = 9\) and also \( (-3)^2 = 9\). This leads us to the concept of taking both positive and negative roots when solving equations through the method of square roots.

Once you understand square roots, grasping the concept of positive and negative roots comes naturally. Whenever you take the square root of both sides in a quadratic equation like \(x^2 = 9\), you must consider that there are two solutions: \(x = \sqrt{9}\) and \(x = -\sqrt{9}\). The reason is that squaring either a positive or a negative number yields the same positive result.

In our exercise, \(x^2 = 9\), the square root of 9 is 3, hence the equation \(x = \sqrt{9}\) yields \(x = 3\). But, it's important to include the negative root, yielding \(x = -3\). Always keep in mind to examine both: the positive root gives you one solution, and the negative root gives you the other. Without considering both roots, you might miss out on valid solutions to the equation.