Factoring General Trinomials

When the leading coefficient is 1, expanding (x - p)(x - q) gives x² - (p+q)x + pq. That means the x-coefficient is just p + q and the constant is pq. So factoring x² + bx + c is a search: find two numbers with that exact sum and product, then drop them into the two binomials. This works because multiplication of binomials always produces this pattern, so matching coefficients backward always works.

When the leading coefficient isn't 1, as in ax² + bx + c, the same sum-and-product idea still drives things, but now you need two numbers that multiply to a times c and add to b. Split the middle term using those two numbers, which turns three terms into four. Group the first two and last two, pull out the common factor from each pair, and you're left with two matching binomial groups you can factor out together. That matching is not an accident: it's guaranteed whenever your two numbers were chosen correctly.

Before doing any of this, always check for a common factor across all three terms first, since factoring that out shrinks the numbers you're hunting for and can reveal a simpler trinomial underneath. This topic sits next to quadratic formula and completing the square work: factoring is faster when it works cleanly, but those other tools cover the cases where no nice integer pair exists.