Question

I would deeply appreciate it if you could help me with my assignment

Answer 1

Given the expression:

$\sqrt{4x}\cdot\sqrt{x+2}$

You need to apply the Property for Radicals that states that the Multiplication of two roots with the same indices is equal to the root of the Product. Then:

$=\sqrt{(4x)(x+2)}$

Apply the Distributive Property inside the square root:

$\begin{gathered} =\sqrt{(4x)(x)+(4x)(2)} \\ \\ =\sqrt{4x^2+8x} \end{gathered}$

By definition, the Radicand (the value inside the square root) must be greater than or equal to zero:

$4x^2+8x\ge0$

Solving for "x", you get:

- Case 1

$\begin{gathered} x\ge0 \\ \\ x+2\ge0\Rightarrow x\ge-2 \end{gathered}$

- Case 2:

$\begin{gathered} x\leq0 \\ \\ x+2\leq0\Rightarrow x\leq-2 \end{gathered}$

By determining the Intersection, you get that the solution is:

$\begin{gathered} x\ge0 \\ x\leq-2 \end{gathered}$

Hence, the answer is: Option B and Option D.