Factoring Quadratic Expressions

Before diving into complex patterns, we always look for the Greatest Common Factor (GCF). Think of this as the 'reverse distributive property.' If every term in your expression shares a common number or variable, you can 'pull it out' to simplify the rest of the problem. For example, in the expression $4x^2 + 8x$, both terms can be divided by $4x$. By factoring out the GCF, we rewrite it as $4x(x + 2)$. This makes the remaining expression much easier to handle. Always check for a GCF first—it is the most common 'hidden' step that makes difficult problems simple.

When you have a trinomial like $x^2 + bx + c$, your goal is to break it into two binomials: $(x + m)(x + n)$. To find $m$ and $n$, you need to find two numbers that multiply to equal $c$ (the product) and add to equal $b$ (the sum). This is often called the 'X-method' or 'Diamond method.' If $c$ is positive, your two numbers will have the same sign. If $c$ is negative, your numbers will have opposite signs. This logic helps you narrow down the possibilities quickly.

Sometimes you'll encounter a binomial that looks like $a^2 - b^2$. This is a special pattern called the Difference of Squares. It only works if: 1) There are exactly two terms, 2) Both terms are perfect squares, and 3) They are being subtracted. The pattern is always $(a - b)(a + b)$. For example, $x^2 - 16$ becomes $(x - 4)(x + 4)$. This is one of the fastest factoring methods because it requires no 'sum and product' guessing—just recognize the squares!