Let's solve the equation [tex]\(4 \sqrt{x+2} = -16\)[/tex] step-by-step and determine if the solution is extraneous or not:
1. Starting the equation:
[tex]\[ 4 \sqrt{x+2} = -16 \][/tex]
2. Isolate the square root term:
Divide both sides by 4 to isolate the square root term:
[tex]\[ \sqrt{x+2} = \frac{-16}{4} \][/tex]
[tex]\[ \sqrt{x+2} = -4 \][/tex]
3. Considering the square root property:
Recall that the square root of a number is always non-negative, meaning [tex]\(\sqrt{y} \geq 0\)[/tex] for any [tex]\(y \geq 0\)[/tex].
4. Check for real solutions:
Since [tex]\(\sqrt{x+2} = -4\)[/tex] implies that the square root, which is always non-negative (zero or positive), must equal a negative value [tex]\(-4\)[/tex]. This is impossible because the square root of a real number cannot be negative.
5. Conclusion:
There are no real solutions to this equation. This means the equation [tex]\(4 \sqrt{x+2} = -16\)[/tex] does not hold true for any real value of [tex]\(x\)[/tex].
Hence, there are no solutions at all, real or extraneous. As a result, we do not encounter any of the suggested values ([tex]\(x = 14\)[/tex] or [tex]\(x = 2\)[/tex]) in the process described.
Final answer: There are no solutions. None of the given options apply.
