Question

What is the domain of `sqrt(2x^2+x-6)`?

Answer 1

The domain is `x in ] -oo,-2 ] uu [3/2, oo[`

Explanation:

We factorise

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

Then,

`2x^2+x-6>=0`

`(2x-3)(x+2)>=0`

Let `f(x)=(2x-3)(x+2)`

We have to do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaa)` `3/2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `2x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+`

Therefore,

As `f(x)>=0`

The domain is `x in ] -oo,-2 ] uu [3/2, oo[`