Question

find the value of the expression below at x = -6. show the steps in your calculation and simplify your answer.\\(\sqrt{\frac{x^2 - 11}{|x + 2|}} = \\)\
what goes in the ( )?\
\\(= \\) (a) \\(\sqrt{\frac{(-6)^2 - 11}{|(-6) + 2|}}\\) or (b) \\(\sqrt{\frac{(6)^2 - 11}{|(6) + 2|}}\\)\
\\(= \\) (a) \\(\sqrt{\frac{36 - 11}{|8|}}\\) or (b) \\(\sqrt{\frac{36 - 11}{|-4|}}\\)\
\\(= \\) (a) \\(\sqrt{\frac{25}{4}}\\) or (b) \\(\sqrt{\frac{25}{8}}\\)\
\\(= \\) (a) \\(\frac{5}{2}\\) or (b) \\(\frac{5\sqrt{8}}{8}\\)

Explanation:

Step1: Substitute $x=-6$ into expression

$$\sqrt{\frac{(-6)^2 - 11}{|(-6)+2|}}$$

Step2: Calculate numerator and denominator

Numerator: $(-6)^2 -11 = 36-11=25$
Denominator: $|(-6)+2|=|-4|=4$
Expression becomes: $\sqrt{\frac{25}{4}}$

Step3: Simplify the square root

$$\sqrt{\frac{25}{4}}=\frac{\sqrt{25}}{\sqrt{4}}=\frac{5}{2}$$

Answer:

The value of the expression at $x=-6$ is $\frac{5}{2}$.
The blank in the parentheses is $-6$.
The correct options are:

1. (a) $\sqrt{\frac{(-6)^2 - 11}{|(-6)+2|}}$
2. (b) $\sqrt{\frac{36-11}{|-4|}}$
3. (a) $\sqrt{\frac{25}{4}}$
4. (a) $\frac{5}{2}$