Question

How do you factor `2x^2 + 18x + 36`?

Answer 1

`2x^2+18x+36 = 2(x+3)(x+6)`

Explanation:

The rule to factorise any quadratic is to find two numbers such that

`"product" = x^2 " coefficient "xx" constant coefficient"`
`"sum" \ \ \ \ \ \ = x " coefficient"`

So for `2x^2+18x+36` first we note that there is a common factor of `2` which we can immediately factor out to give:

`2x^2+18x+36 = 2(x^2+9x+18)`

we seek two numbers such that

`"product" = (1)*(18) = 18`
`"sum" \ \ \ \ \ \ = 9`

So we look at the factors of `18`. As the sum is negative and the product is positive then both factors must be positive, We can check every combination of the product factors:

`{: ("factor1", "factor2", "sum"), (1,18,19),(2,9,11),(3,6,9) :}`

So the factors we seek are `color(blue)(3)` and `color(green)(6)`. With practice one can determine the appropriate factors by inspection alone.

Therefore we can factorise the quadratic as follows:

`x^2+9x+18= x^2 color(blue)(+3)x color(green)(+6)x +18`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= x(x+3)+6(x+3)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (x+6)(x+3)`

Similarly if we grouped the factors the other way around we get the same answer:

`x^2+9x+18= x^2 color(green)(+6)x color(blue)(+3)x +18`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= x(x+6)+3(x+6)`
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \= (x+3)(x+6)`

This approach works for all quadratics (assuming it does factorise) , The middle step in the last section can usually be skipped with practice.

Hence,

`2x^2+18x+36 = 2(x+3)(x+6)`