What is the domain of $\sqrt{2 x^{2} + x - 6}$ $sqrt(2x^2+x-6)$ ?

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What is the domain of $\sqrt{2 x^{2} + x - 6}$ $sqrt(2x^2+x-6)$ ?

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Answer 1 · 2017-01-03T08:48:24

The domain is $x \in] - \infty, - 2] \cup [\frac{3}{2}, \infty [$ $x in ] -oo,-2 ] uu [3/2, oo[$

Explanation:

We factorise

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

Then,

$2 x^{2} + x - 6 \geq 0$ $2x^2+x-6>=0$

$(2 x - 3) (x + 2) \geq 0$ $(2x-3)(x+2)>=0$

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

We have to do a sign chart

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a$ $color(white)(aaaa)$ $- 2$ $-2$ $a a a a a$ $color(white)(aaaaa)$ $\frac{3}{2}$ $3/2$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $x + 2$ $x+2$ $a a a a a a$ $color(white)(aaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $2 x - 3$ $2x-3$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $f (x)$ $f(x)$ $a a a a a a a$ $color(white)(aaaaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a a a$ $color(white)(aaaaa)$ $+$ $+$

Therefore,

As $f (x) \geq 0$ $f(x)>=0$

The domain is $x \in] - \infty, - 2] \cup [\frac{3}{2}, \infty [$ $x in ] -oo,-2 ] uu [3/2, oo[$

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What is the domain of $\sqrt{2 x^{2} + x - 6}$ $sqrt(2x^2+x-6)$ ?

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