91Ó°ÊÓ

Factoring quadratics is a critical skill for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The process involves breaking down the quadratic into a product of simpler expressions (linear factors) that, when multiplied together, give the original quadratic. This makes it easier to find the values of \(x\) that satisfy the equation (i.e., where \(f(x) = 0\)). One common technique for factoring is to look for a pair of numbers that both add up to \(b\) and multiply to \(ac\).
For the equation \(x^2 - 9\), you might notice that it is a special form called the difference of squares which factors neatly without needing to find such a pair. Understanding how to recognize these patterns in quadratics is essential for efficient problem solving.

The difference of squares is a special algebraic pattern that takes the form \(a^2 - b^2\) and can be factored into \((a + b)(a - b)\). Recognizing this pattern is useful because it simplifies the process of factoring dramatically. For the quadratic equation given, \(x^2 - 9\), we can identify \(9\) as a perfect square of \(3\), rendering the equation a difference of squares, \(x^2 - 3^2\).
It can then be factored into \((x + 3)(x - 3)\). This pattern recognition is key in algebra, especially when dealing with quadratic equations, because it allows for quick simplification without much trial and error.

When it comes to finding real values of \(x\) that satisfy a quadratic equation, we need to solve for the instances when the equation equals zero (\(f(x) = 0\)). After factoring the quadratic equation, as in the previous steps, we set each factor equal to zero because if the product of two factors is zero, at least one of the factors must also be zero. This is known as the zero-product property.
For the given factored equation \((x + 3)(x - 3) = 0\), we solve for \(x\) by setting each factor equal to zero: \(x + 3 = 0\) and \(x - 3 = 0\). Solving these gives us the real values \(x = -3\) and \(x = 3\). This step is crucial as it provides the actual solutions to the original quadratic equation.

The zero-product property is a key principle in algebra that states if the product of two factors is zero, then at least one of the factors must be zero. This property is used after factoring a quadratic equation to find the solutions for \(x\).
In the context of the exercise, once we factored the equation as \((x - 3)(x + 3) = 0\), we applied the zero-product property. By setting each factor equal to zero: \(x - 3 = 0\) and \(x + 3 = 0\), we could solve for \(x\) and find the real values that satisfy the equation. This property is a cornerstone of solving quadratics because it allows us to break down the problem into simpler, solvable pieces.