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

1 Answer
Jan 3, 2017

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[#