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

Jan 3, 2017

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

Explanation:

We factorise

$2 {x}^{2} + x - 6 = \left(2 x - 3\right) \left(x + 2\right)$

Then,

$2 {x}^{2} + x - 6 \ge 0$

$\left(2 x - 3\right) \left(x + 2\right) \ge 0$

Let $f \left(x\right) = \left(2 x - 3\right) \left(x + 2\right)$

We have to do a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a}$$\frac{3}{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 2$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$2 x - 3$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$+$

Therefore,

As $f \left(x\right) \ge 0$

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