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Factoring quadratic equations is a common method for solving these types of equations. When a quadratic equation is presented in the standard form, such as \(ax^2 + bx + c = 0\), the goal is to rewrite or "factor" it as a product of two binomials. This makes it easier to find the solutions for the quadratic equation. The process involves finding two numbers that both add to the coefficient of \(x\) (in this case, \(b\)) and multiply to the constant term \(a\cdot c\).
For instance, consider the given equation \(6x^2 + 13x + 6 = 0\). We need two numbers that add up to 13 (the middle term) and multiply to 18 (the product of 6 and 3). The correct pair is 9 and 2 because 9 + 2 = 13 and 9 \times 2 = 18. Thus, the equation is factored as \((2x + 3)(3x + 2) = 0\).
This factoring technique transforms the quadratic equation into a simpler form where the zero factor property can be applied, leading us to the solutions of the original equation.

The zero factor property is an essential principle used in solving quadratic equations after they are factored. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. When you have an equation in the form \(ab = 0\), then either \(a = 0\) or \(b = 0\).
In the context of our factored quadratic equation \((2x + 3)(3x + 2) = 0\), we apply the zero factor property by setting each binomial equal to zero:

• Set \(2x + 3 = 0\) and solve for \(x\).
• Set \(3x + 2 = 0\) and solve for \(x\).

This approach simplifies solving the equation because once you factor the equation, you deal with two simple linear equations instead of the original quadratic form. This concept is crucial for reducing complex polynomial equations to more easily manageable parts.

Solving quadratic equations involves finding the values of \(x\) that make the equation true. There are several methods to solve quadratic equations: factoring, completing the square, using the quadratic formula, and graphing. Factoring is often the easiest and quickest method when applicable.
In the exercise provided, we used factoring because the quadratic equation could be rewritten as the product \((2x + 3)(3x + 2) = 0\). Once factored, we utilized the zero factor property to find the solutions. Setting each factor equal to zero gave us the equations \(2x + 3 = 0\) and \(3x + 2 = 0\). Solving these yields the solutions \(x = -\frac{3}{2}\) and \(x = -\frac{2}{3}\).
These solutions are the points where the parabola, represented by the original quadratic equation, crosses the x-axis. Each method has its place, but factoring is typically preferred when both a quick solution is available and when dealing with integers or simple fractions.