text: # How do you solve sqrt(x + 7) = x - 5?
  
  Sep 7, 2016
  
  $x = 9$
  
  #### Explanation:
  
  Given:
  
  $\sqrt{x + 7} = x - 5$
  
  Square both sides of the equation to get:
  
  $x + 7 = {\left(x - 5\right)}^{2} = {x}^{2} - 10 x + 25$
  
  Subtracting $x + 7$ from both ends, we get:
  
  $0 = {x}^{2} - 11 x - 18 = \left(x - 2\right) \left(x - 9\right)$
  
  So the solutions of this derived quadratic equation are $x = 2$ and $x = 9$.
  
  Any solutions of the original equation must be solutions of the derived quadratic equation, so $x = 2$ and $x = 9$ are the only possible solutions of the original equation.
  
  However, note that squaring is not a one to one function, so solutions of the derived equation may not be solutions of the original one.
  
  In fact, we find:
  
  $\sqrt{\textcolor{red}{2} + 7} = \sqrt{9} = 3 \ne - 3 = \textcolor{red}{2} - 5$
  
  So $x = 2$ is not a solution of the original equation.
  
  On the other hand, we find:
  
  $\sqrt{\textcolor{red}{9} + 7} = \sqrt{16} = 4 = \textcolor{red}{9} - 5$
  
  So $x = 9$ is a solution of the original equation.
explanation: The document contains a math question and a detailed answer to the question. The question is about solving the equation \(\sqrt{x + 7} = x - 5\), and the answer provides a step-by-step solution, including the verification of the solutions.
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: # How do you solve sqrt(x + 7) = x - 5?
  
  Sep 7, 2016
  
  $x = 9$
  
  #### Explanation:
  
  Given:
  
  $\sqrt{x + 7} = x - 5$
  
  Square both sides of the equation to get:
  
  $x + 7 = {\left(x - 5\right)}^{2} = {x}^{2} - 10 x + 25$
  
  Subtracting $x + 7$ from both ends, we get:
  
  $0 = {x}^{2} - 11 x - 18 = \left(x - 2\right) \left(x - 9\right)$
  
  So the solutions of this derived quadratic equation are $x = 2$ and $x = 9$.
  
  Any solutions of the original equation must be solutions of the derived quadratic equation, so $x = 2$ and $x = 9$ are the only possible solutions of the original equation.
  
  However, note that squaring is not a one to one function, so solutions of the derived equation may not be solutions of the original one.
  
  In fact, we find:
  
  $\sqrt{\textcolor{red}{2} + 7} = \sqrt{9} = 3 \ne - 3 = \textcolor{red}{2} - 5$
  
  So $x = 2$ is not a solution of the original equation.
  
  On the other hand, we find:
  
  $\sqrt{\textcolor{red}{9} + 7} = \sqrt{16} = 4 = \textcolor{red}{9} - 5$
  
  So $x = 9$ is a solution of the original equation.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer to the question. The question is about solving the equation \(\sqrt{x + 7} = x - 5\), and the answer provides a step-by-step solution, including the verification of the solutions.
  
  Label: question-and-answer