Question

How do you factor `2x^2+15x+18`?

Answer 1

`(2x+3)(x+6)`

Explanation:

`2x^2+15x+18 = 2x^2+12x+3x+18 = 2x(x+6)+3(x+6)=(2x+3)(x+6)`[Ans]

Answer 2

`color(magenta)2x^2 + 15x color(lime)+ color(purple)18`

Find factors of `2 and 18` which ADD (because of `color(lime)(+)` 18) to make `15.`

The signs will be the same (because of the `color(lime)(+)18)`, they are both positive (because of the +15).

`2" "18`
`darr" "darr`

`color(red)(2)" "color(blue)(3) " cross multiply and ADD the products"`
`color(blue)(1)" "color(red)(6)`

`color(red)(2)" "color(blue)(3rArr 1 xx3 = 3)`
`color(blue)(1)" "color(red)(6 rArr 2 xx6 = ul12`
`color(white)(xxxxxxxx.xxxxx)15`

The top line gives one of the factors and the bottom line gives the other factor.

`(2x +3)(x + 6)`

Answer 3

`(2x + 3)(x + 6)`

Explanation:

Use the new AC method (Socratic Search)

`y = 2x^2 + 15 x + 18 = 2(x + p)(x + q)`

Converted trinomial: `y' = x^2 + 15x + 36 = (x + p')(x + q').`

Find `p' and q'` knowing sum `(b = 15)` and product `(ac = 36).`

They are: `p' = 3 and q' = 12.`
Back to original `y rarr p = (p')/a = 3/2`, and `q = (q')/a = 12/2 = 6`.

Factored form: `2(x + 3/2)(x + 6) = (2x + 3)(x + 6)`