When a quadratic trinomial x squared plus bx plus c factors as (x+p)(x+q), the constant c equals the product of p and q. Learn to find the missing c from a factored form or known roots, and how it connects to finding the missing b.
Finding the unknown c in x² + bx + c
A quadratic trinomial in the form x² + bx + c factors as (x + p)(x + q) when two numbers p and q multiply to give c and add to give b. If you're given the factored form, or the two numbers p and q, finding c is just one multiplication: c = p × q.
From the factored form (x+2)(x+5), c is the product of the constants: 2 × 5 = 10.
Worked example
Given the factored form (x + 2)(x + 5), find c in x² + bx + c. The constant term comes from multiplying the two numbers: c = 2 × 5 = 10. (The coefficient b comes from adding them: b = 2 + 5 = 7.) So x² + 7x + 10 = (x + 2)(x + 5).
Working backward from roots
If you're given the two roots of the equation instead of a factored expression, the same idea applies: c is still the product of the two root-related numbers. This mirrors factoring trinomials in reverse — instead of factoring x² + bx + c, you're building it from known factors.
Why this matters
This skill connects finding missing coefficients to the broader pattern used in finding the unknown b and general polynomial factoring: once you know a trinomial's factored form, both b and c follow directly from the two numbers inside the parentheses.