Question

How do you factor the trinomial `3x^2 + 11x + 6`?

Answer 1

`=(x+3)(3x+2)`

Explanation:

`3x^2+11x+6` => using the AC method (see below):
`=3x^2+9x+2x+6`
`=3x(x+3)+2(x+3)`
`=(x+3)(3x+2)`

Factoring quadratics:
A quadratic equation in general is defined as:
`Ax^2 + Bx + C = 0`
Example (easy case when A = 1)
`x^2 + 5x + 6` => look for two numbers that add to +5, multiply to + 6 i.e. (3 & 2)
`=(x + 3)(x + 2)`

The AC Method:
What can we do when the leading coefficient is not 1?
We use an extension of factoring by grouping called the AC method.
Step by Step method for factoring `Ax^2 + Bx + C` :
Step 1. Multiply together AC and list the factors of AC.
Step 2. Find a pair that adds to B. If you can't find such pair the quadratic is a prime and does not factor.
Step 3. Rewrite the middle term as a sum of terms whose coefficients are the chosen pair.
Step 4. Factor by grouping.
Remember you should always first pull out the GCF.

Examples:
1) `2x^2 + 5x - 25`
AC = (2)(-25) = -50
the pairs are:
(1,-50), (-1,50), (2,-25), (-2,25), (5,-10) and (-5,10).
We see that: -5 + 10 = 5 hence we choose the pair (-5,10)
We write:
`2x^2 - 5x + 10x - 25`
`= (2x^2 - 5x) + (10x - 25)`
`= x(2x - 5) + 5 (2x - 5)`
`= (x + 5) (2x - 5)`

2) `9x^2- 49x - 30`
AC = (9)(-30) = -270 => Hint : find the pairs that add to -49 i.e, (-54 , 5):
`= 9x^2 - 54x + 5x - 30`
`= 9x(x - 6) + 5(x - 6)`
`= (x - 6)(9x + 5)`

I hope this helps.