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Factoring quadratics is a fundamental technique in algebra, especially when dealing with quadratic equations. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \). The main goal in factoring is to express this equation as a product of two simple expressions. This helps in finding the solutions easily.

Consider the equation \( f(x) = x^2 - 9 \). This equation is already in the standard quadratic form where \( a = 1 \), \( b = 0 \), and \( c = -9 \). Our job is to break it down into two binomials. Recognize that \( x^2 - 9 \) is a special form known as the difference of squares. The difference of squares formula is given by \( a^2 - b^2 = (a-b)(a+b) \).

So, for \( x^2 - 9 \), you can rewrite it as \((x - 3)(x + 3) = 0 \). This shows that the quadratic can indeed be expressed as the product of two linear factors. Factoring like this makes it much easier to solve the equation and find the roots.

Once a quadratic equation is factored correctly, solving it becomes straightforward. Solving an equation essentially means finding the value(s) of the variable that make the equation true. In terms of a quadratic equation, these solutions are often referred to as roots.

From the factorized form \((x - 3)(x + 3) = 0 \), the task is to solve each factor individually. The Zero-Product Property tells us that if the product of two numbers is zero, then at least one of the numbers must be zero. This helps us conclude:

• For \( (x-3) = 0 \), add 3 to both sides to get \( x = 3 \).
  
• For \( (x+3) = 0 \), subtract 3 from both sides to get \( x = -3 \).
  

These solutions \( x = 3 \) and \( x = -3 \) make the original equation true, meaning we've found the roots through solving the equation after factoring it.

The roots of a function are the values of \( x \) that make the function equal to zero. In other words, they are the solutions to the equation \( f(x) = 0 \). For a quadratic function, there can be up to two distinct roots.

In our exercise, the function was \( f(x) = x^2 - 9 \). After factoring and solving, we found the roots \( x = 3 \) and \( x = -3 \). These roots indicate the points where the graph of the quadratic function intersects the x-axis.

Understanding roots is crucial for graphing the function and analyzing its behavior. Whether a function crosses or just touches the x-axis at these points tells us a lot about its nature and shape. The term "roots" may also be referred to as "solutions" or "zeros" of the function.