text: +0
  
  # math
  
  0
  160
  4
  
  Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.
  
  May 4, 2021
  
  #1
  +32741
  +3
  
  Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.
  
  (where I have distance from origin, I should have put distance squared from origin!)
  
  May 4, 2021
  edited by Alan  May 4, 2021
  #2
  +2266
  0
  
  Origin is: (0, 0)
  
  Point in parabola: (x, y)
  
  Distance: sqrt(x^2 + y^2)
  
  sqrt(x^2 + y^2)
  
  sqrt(x^2 + (x^2 - 2)^2)
  
  sqrt(x^2 + x^4 - 4x^2 + 4)
  
  sqrt(x^4 - 3x^2 + 4)
  
  Now, we have to find the smallest positive value of x^4 - 3x^2 + 4.
  
  Not quite sure how to do that.
  
  =^._.^=
  
  May 4, 2021
  #3
  +1
  
  Can use calculas as in #1 above, or altermatively completing the square works.
  
  $$\displaystyle x^{4}-3x^{2}+4 = x^{4}-3x^{2}+(9/4) + 4 -(9/4)\\ = (x^{2}-3/2)^{2}+7/4,$$
  
  so the minimum distance is $$\displaystyle \sqrt{7/4}$$ from either $$\displaystyle x=\pm\sqrt{3/2}.$$
  
  May 4, 2021
  #4
  +2266
  0
  
  Ohhh, I can't believe I didn't think of completing the square.
  
  Nice job :))
  
  =^._.^=
  
  catmg  May 4, 2021
explanation: The document contains a math question and multiple answers to the question. The question is about finding the smallest distance between the origin and a point on the parabola \( y = x^2 - 2 \). There are three responses providing different approaches to solving the problem.
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: +0
  
  # math
  
  0
  160
  4
  
  Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.
  
  May 4, 2021
  
  #1
  +32741
  +3
  
  Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.
  
  (where I have distance from origin, I should have put distance squared from origin!)
  
  May 4, 2021
  edited by Alan  May 4, 2021
  #2
  +2266
  0
  
  Origin is: (0, 0)
  
  Point in parabola: (x, y)
  
  Distance: sqrt(x^2 + y^2)
  
  sqrt(x^2 + y^2)
  
  sqrt(x^2 + (x^2 - 2)^2)
  
  sqrt(x^2 + x^4 - 4x^2 + 4)
  
  sqrt(x^4 - 3x^2 + 4)
  
  Now, we have to find the smallest positive value of x^4 - 3x^2 + 4.
  
  Not quite sure how to do that.
  
  =^._.^=
  
  May 4, 2021
  #3
  +1
  
  Can use calculas as in #1 above, or altermatively completing the square works.
  
  $$\displaystyle x^{4}-3x^{2}+4 = x^{4}-3x^{2}+(9/4) + 4 -(9/4)\\ = (x^{2}-3/2)^{2}+7/4,$$
  
  so the minimum distance is $$\displaystyle \sqrt{7/4}$$ from either $$\displaystyle x=\pm\sqrt{3/2}.$$
  
  May 4, 2021
  #4
  +2266
  0
  
  Ohhh, I can't believe I didn't think of completing the square.
  
  Nice job :))
  
  =^._.^=
  
  catmg  May 4, 2021
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and multiple answers to the question. The question is about finding the smallest distance between the origin and a point on the parabola \( y = x^2 - 2 \). There are three responses providing different approaches to solving the problem.
  
  Label: multiple-question-and-answer