Question

How do you factor `6x^2+19x+3`?

Answer 1

Use the new AC Method to factor.

`f(x) = 6x^2 + 19x + 3 = (x - p)(x - q)`
Converted trinomial:`f'(x) = x^2 + 19x + 18 = (x - p')(x - q').`
Compose factor pairs of (a.c) = 18. Proceed (1, 18) = 19 = b. Then p' = 1 and q' = 18. We get: `p = (p')/a = 1/6`and `q = (q')/a = 18 /6 = 3`.
Factored form:`f(x) = (x + 1/6)(x + 3) = (6x + 1)(x + 3)`
Check: Develop`f(x) = 6x^2 + 18x + x + 3 = 6x^2 + 19x + 3`. Correct

Reminder about the Rule of Signs.
a. When a and c have same signs, compose factor pairs of c, or (a.c), with all positive numbers.
Example: f(x) = x^2 + 51x + 98 = (x - p)(x - q). Compose factor pairs of (a.c = 98). a and are both positive. Proceed: (1, 98)(2, 49). This last sum is 51 = b. Then p = 2 and q = 49.

b. When a and c have different signs, compose factor pairs of c, or (a.c), with all first numbers being negative.
Example. f(x) = x^2 - 11x - 42 = (x - p)(x - q). Compose factor pairs of (c = -42). Proceed: (-1, 42)(-2, 21)(-3, 14). This last sum is 11 = -b. Then the 2 real roots are -3 and 14.