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Factoring quadratics is a method used to find the roots of quadratic equations by expressing the equation in terms of its factors. When a quadratic equation is of the form \(ax^2 + bx + c = 0\), the first step is to look for two numbers that multiply to the constant term \(c\), but also add to the linear term \(b\). This process allows us to rewrite the quadratic as a product of two binomials.
In the example \(x^2 - 8x + 12 = 0\), we can set it up like this: the two numbers that add to -8 and multiply to 12 are -2 and -6, hence we factor it into \((x - 2)(x - 6) = 0\).
Factoring is particularly useful as it reduces a quadratic equation into simpler linear equations. This makes finding the solution much more straightforward.

The roots of a quadratic are the values of \(x\) that make the equation equal to zero. Finding the roots is essentially the solution to the quadratic equation. These roots are also known as the 'zeroes' or 'x-intercepts' of the quadratic function.
Once we have factored the quadratic equation, such as \(x^2 - 8x + 12\) into \((x - 2)(x - 6) = 0\), we set each factor equal to zero and solve for \(x\). Here, by setting \(x - 2 = 0\) and \(x - 6 = 0\), we find the roots \(x = 2\) and \(x = 6\).
These values are where the graph of the quadratic equation intersects the x-axis. Understanding the roots helps in graphing the equation and in various real-world applications.

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. This can be achieved using multiple methods like factoring, completing the square, or utilizing the quadratic formula. For simpler equations, factoring is often the quickest method.
The step-by-step method requires you to:

• Write down the quadratic equation.
• Factor it into two binomials.
• Set each binomial equal to zero to find the solutions.

For the equation \(x^2 - 8x + 12 = 0\), after factoring into \((x - 2)(x - 6) = 0\), solving for \(x\) gives us the solutions \(x = 2\) and \(x = 6\).
While factoring is suitable for quadratics that can be neatly broken down, other methods may be required for more complex equations. Thus, understanding the characteristic of the equation helps in choosing the most efficient method.