# If $y=\sqrt {x^2+6x+8}$, then the value of $\sqrt {1+iy}+\sqrt {1-iy}$= ?

$\begin{array}{1 1}(A)\sqrt {2(x+4)} \\ (B) \sqrt {(x+4)} \\ (C) \sqrt {2(x+3)} \\(D) \sqrt {(x+3)} \end{array}$

Let $z=\sqrt {1+iy}+\sqrt{1-iy}$
$\Rightarrow\:z^2=1+iy+1-iy+2\sqrt{(1+iy)(1-iy)}$
$=2+2\sqrt{1+y^2}$
Given: $y=\sqrt {x^2+6x+8}$
$\therefore\:z^2=2+2\sqrt {1+x^2+6x+8}$
$=2+2\sqrt {(x+3)^2}=2(1+x+3)=2(x+4)$
$\Rightarrow\:z=\sqrt {2(x+4)}$
The text passage contains a mathematical question involving a complex expression and its solution, with multiple choice options provided. The solution to the problem is also given in the text.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# If $y=\sqrt {x^2+6x+8}$, then the value of $\sqrt {1+iy}+\sqrt {1-iy}$= ?

$\begin{array}{1 1}(A)\sqrt {2(x+4)} \\ (B) \sqrt {(x+4)} \\ (C) \sqrt {2(x+3)} \\(D) \sqrt {(x+3)} \end{array}$

Let $z=\sqrt {1+iy}+\sqrt{1-iy}$
$\Rightarrow\:z^2=1+iy+1-iy+2\sqrt{(1+iy)(1-iy)}$
$=2+2\sqrt{1+y^2}$
Given: $y=\sqrt {x^2+6x+8}$
$\therefore\:z^2=2+2\sqrt {1+x^2+6x+8}$
$=2+2\sqrt {(x+3)^2}=2(1+x+3)=2(x+4)$
$\Rightarrow\:z=\sqrt {2(x+4)}$
user
The text passage contains a mathematical question involving a complex expression and its solution, with multiple choice options provided. The solution to the problem is also given in the text.

TYPE: SINGLE_QUESTION_WITH_ANSWER