To determine the solution for the equation [tex]\(4 \sqrt{x + 2} = -16\)[/tex] and its validity, let's proceed through the steps clearly and logically.
### Step-by-Step Solution:
1. Understand the equation:
The equation we are trying to solve is:
[tex]\[
4 \sqrt{x + 2} = -16
\][/tex]
2. Isolate the square root term:
Divide both sides of the equation by 4 to isolate the square root:
[tex]\[
\sqrt{x + 2} = \frac{-16}{4}
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
\sqrt{x + 2} = -4
\][/tex]
3. Analyze the situation:
Recall that the square root of a number represents its non-negative principal root. This means [tex]\(\sqrt{x + 2}\)[/tex] must be non-negative (i.e., [tex]\(\sqrt{x + 2} \geq 0\)[/tex]). However, the equation states [tex]\(\sqrt{x + 2} = -4\)[/tex], which is a contradiction since a square root cannot be a negative number.
4. Conclusion:
Because the equation [tex]\(\sqrt{x + 2} = -4\)[/tex] is impossible to satisfy, there are no real values of [tex]\(x\)[/tex] that will solve the original equation.
### Determination of Solutions:
Since there is no value of [tex]\(x\)[/tex] that can satisfy the given equation, it means that the equation has no solution. Therefore, we do not need to check for extraneous solutions because there are no solutions to begin with.
### Final Answer:
Given the provided options:
- [tex]\(x = 14\)[/tex]; extraneous
- [tex]\(x = 14\)[/tex]; not extraneous
- [tex]\(x = 2\)[/tex]; extraneous
- [tex]\(x = 2\)[/tex]; not extraneous
The correct choice is:
[tex]\[
\boxed{\text{None of the given options are correct since the equation has no solution.}}
\][/tex]
To validate, our result indicates that there are no solutions to the equation [tex]\(4 \sqrt{x + 2} = -16\)[/tex], and thus, all provided options do not represent possible solutions.
