text: # 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: $y=\sqrt {x^2+6x+8}$
  $\therefore\:z^2=2+2\sqrt {1+x^2+6x+8}$
  $=2+2\sqrt {(x+3)^2}=2(1+x+3)=2(x+4)$
  $\Rightarrow\:z=\sqrt {2(x+4)}$
explanation: The document contains a math question and its corresponding solution. The question asks for the value of \(\sqrt{1+iy} + \sqrt{1-iy}\) given \(y = \sqrt{x^2 + 6x + 8}\). The solution is provided, leading to the answer \(\sqrt{2(x+4)}\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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: $y=\sqrt {x^2+6x+8}$
  $\therefore\:z^2=2+2\sqrt {1+x^2+6x+8}$
  $=2+2\sqrt {(x+3)^2}=2(1+x+3)=2(x+4)$
  $\Rightarrow\:z=\sqrt {2(x+4)}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding solution. The question asks for the value of \(\sqrt{1+iy} + \sqrt{1-iy}\) given \(y = \sqrt{x^2 + 6x + 8}\). The solution is provided, leading to the answer \(\sqrt{2(x+4)}\).
  
  Label: question-and-answer