When a quadratic trinomial x squared plus bx plus c factors as (x+p)(x+q), the coefficient b equals the sum of p and q. Learn to find the missing b from a factored form, from known factor pairs of c, and how it relates to finding the missing c.
Finding the unknown b in x² + bx + c
A quadratic trinomial x² + bx + c factors as (x + p)(x + q) when p and q multiply to give c and add to give b. If you're given c along with the two factors, or the factored form directly, finding b just means adding the two numbers.
From the factored form (x+3)(x+6), b is the sum of the constants: 3 + 6 = 9.
Worked example
Given the factored form (x + 3)(x + 6), find b in x² + bx + 18. Add the two numbers inside the parentheses: b = 3 + 6 = 9. (Multiplying them instead gives c: 3 × 6 = 18, confirming the trinomial.) So x² + 9x + 18 = (x + 3)(x + 6).
Finding factors when only c is known
If you're only given c, list pairs of numbers that multiply to c, then add each pair to see which gives a matching b. For c = 18, the pairs are (1,18), (2,9), and (3,6) — each pair adds to a different possible b.
How this connects
This is the mirror skill to finding the unknown c: both come from the same pair of numbers, just combined differently (sum for b, product for c). Together they're the foundation of factoring trinomials in general.