Question

How to simplify (2x^2+10x-12)/(x^2+x-6) ? Thank you!

Answer 1

`(2(x+6)(x-1))/((x+3)(x-2))`

Explanation:

`"factorise numerator/denominator and cancel any common"`
`"factors"`

`color(blue)"factor numerator"`

`rArr2(x^2+5x-6)`

`"the factors of - 6 which sum to + 5 are + 6 and - 1"`

`=2(x+6)(x-1)larr"numerator factored"`

`color(blue)"factor denominator"`

`"the factors of - 6 which sum to + 1 are + 3 and - 2"`

`rArrx^2+x-6=(x+3)(x-2)`

`rArr(2(x+6)(x-1))/((x+3)(x-2))`

`"there is no cancelling between factors"`

Answer 2

`(2x^2+10x-12)/(x^2+x-6)=[2(x+6)(x-1)]/[(x+3)(x-2)]`

Explanation:

We need to factorise the numerator and the denominator.

Let's start with the denominator, which is the easiest to work out:
We would like `x^2+x-6` on the form `(x+a)(x+b)`
If we multiply it out, we get
`(x+a)(x+b)=x^2+(a+b)x+ab=x^2+x-6`
Now `-6= -(2*3)`.
In addition `a+b=1`
Therefore `a=3, b=-2`
Therefore the numerator would be `(x+3)(x-2)`

Denominator:
`2x^2+10x-12=2(x^2+5x-6)`
Here `a+b=5, a*b=-6`
The only way this works out, is if a=6, b=-1.
Therefore the denominator would be
`2x^2+10x-12=(x+6)(x-1)`

So our expression can be simplified to

`(2x^2+10x-12)/(x^2+x-6)=[2(x+6)(x-1)]/[(x+3)(x-2)]`

Regretably the numerator and the denominator do not share any factor, so we are not able to simplify any more.