To find the solution of the equation [tex]\( 4 \sqrt{2 + 2} = -1.5 \)[/tex] and determine if it is an extraneous solution, follow these steps:
1. Simplify Inside the Square Root:
First, simplify the expression inside the square root:
[tex]\[
2 + 2 = 4
\][/tex]
2. Calculate the Square Root:
Now, take the square root of the simplified value:
[tex]\[
\sqrt{4} = 2
\][/tex]
3. Multiply by 4:
Next, multiply this result by 4, as per the left-hand side of the equation:
[tex]\[
4 \times 2 = 8
\][/tex]
4. Compare with the Right-Hand Side:
Now compare the value obtained with the right-hand side of the equation:
[tex]\[
8 \neq -1.5
\][/tex]
Since the left-hand side (8) is not equal to the right-hand side (-1.5), it shows that there is no valid solution to the equation [tex]\( 4 \sqrt{2 + 2} = -1.5 \)[/tex].
Therefore, the given equation does not have a valid solution, and the statement that [tex]\( x = 14 \)[/tex] is incorrect. Furthermore, the comparison shows that the equation represents an impossible scenario, thus the notion of an extraneous solution is confirmed.
So, the solution is indeed extraneous, as the left-hand side and the right-hand side of the equation are not equal.
