text: What are all solutions of equation f(-3x)=0 if the roots of equation f(x)=0 are -1 and -2?
  
  ken5000 | High School Teacher | (Level 1) Adjunct Educator
  
  Posted on
  
  If -1 and 2 are roots, we can write:
  
  x = -1 and x = 2.
  
  Then x + 1 = 0 and x - 2 = 0
  
  So the factors are x + 1 and x + 2.
  
  And the simplest version of the original function would have been f(x) = (x + 1)(x + 2).
  
  Replacing x with -3x gives us:
  
  f(-3x) = (-3x + 1)(-3x + 2).
  
  Setting these factors equal to 0 gives us:
  
  -3x + 1 = 0 or -3x + 2 = 0
  
  -3x = -1 or -3x = -2
  
  x = 1/3 or x = 2/3
  
  The solution set is:  {1/3, 2/3}.
  
  justaguide | College Teacher | (Level 2) Distinguished Educator
  
  Posted on
  
  The roots of f(x) = 0 are -1 and -2. We can write f(x) = (x + 1)(x + 2)
  
  f(-3x) = 0
  
  => (-3x + 1)(-3x + 2) = 0
  
  The roots of this are x = 1/3 and x = 2/3
  
  The solutions of the equation are x = 1/3 and x = 2/3
  
  giorgiana1976 | College Teacher | (Level 3) Valedictorian
  
  Posted on
  
  Since the equation f(x) = 0 has two roots, then the equation is a quadratic:
  
  f(x) = (x+1)(x+2)
  
  f(x) = x^2 + 3x + 2
  
  f(-3x) = (-3x)^2 + 3*(-3x) + 2
  
  f(-3x) = 9x^2 - 9x + 2
  
  x1 = [9+sqrt(81 - 72)]/18
  
  x1 = (9+3)/18
  
  x1 = 12/18
  
  x1 = 2/3
  
  x2 = (9-3)/18
  
  x2 = 6/18
  
  x2 = 1/3
  
  Another way to solve the problem is to consider that any root of an equation, substituted within equation, verifies it.
  
  f(-1) = 0
  
  But f(-3x) = 0
  
  f(-3x) = f(-3*(1/3)) = f(-1) = 0 => x = 1/3
  
  f(-3x) = f(-3*(2/3)) = f(-2) = 0 => x = 2/3
  
  Therefore, the solutions of the quadratic equation f(-3x)=0 are {1/3 ; 2/3}.
explanation: The document contains a math question and multiple answers provided by different educators. The question asks for the solutions of the equation \( f(-3x) = 0 \) given that the roots of \( f(x) = 0 \) are -1 and -2. Each educator provides a step-by-step solution, leading to the same answer: the solutions are \( x = \frac{1}{3} \) and \( x = \frac{2}{3} \).
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: What are all solutions of equation f(-3x)=0 if the roots of equation f(x)=0 are -1 and -2?
  
  ken5000 | High School Teacher | (Level 1) Adjunct Educator
  
  Posted on
  
  If -1 and 2 are roots, we can write:
  
  x = -1 and x = 2.
  
  Then x + 1 = 0 and x - 2 = 0
  
  So the factors are x + 1 and x + 2.
  
  And the simplest version of the original function would have been f(x) = (x + 1)(x + 2).
  
  Replacing x with -3x gives us:
  
  f(-3x) = (-3x + 1)(-3x + 2).
  
  Setting these factors equal to 0 gives us:
  
  -3x + 1 = 0 or -3x + 2 = 0
  
  -3x = -1 or -3x = -2
  
  x = 1/3 or x = 2/3
  
  The solution set is:  {1/3, 2/3}.
  
  justaguide | College Teacher | (Level 2) Distinguished Educator
  
  Posted on
  
  The roots of f(x) = 0 are -1 and -2. We can write f(x) = (x + 1)(x + 2)
  
  f(-3x) = 0
  
  => (-3x + 1)(-3x + 2) = 0
  
  The roots of this are x = 1/3 and x = 2/3
  
  The solutions of the equation are x = 1/3 and x = 2/3
  
  giorgiana1976 | College Teacher | (Level 3) Valedictorian
  
  Posted on
  
  Since the equation f(x) = 0 has two roots, then the equation is a quadratic:
  
  f(x) = (x+1)(x+2)
  
  f(x) = x^2 + 3x + 2
  
  f(-3x) = (-3x)^2 + 3*(-3x) + 2
  
  f(-3x) = 9x^2 - 9x + 2
  
  x1 = [9+sqrt(81 - 72)]/18
  
  x1 = (9+3)/18
  
  x1 = 12/18
  
  x1 = 2/3
  
  x2 = (9-3)/18
  
  x2 = 6/18
  
  x2 = 1/3
  
  Another way to solve the problem is to consider that any root of an equation, substituted within equation, verifies it.
  
  f(-1) = 0
  
  But f(-3x) = 0
  
  f(-3x) = f(-3*(1/3)) = f(-1) = 0 => x = 1/3
  
  f(-3x) = f(-3*(2/3)) = f(-2) = 0 => x = 2/3
  
  Therefore, the solutions of the quadratic equation f(-3x)=0 are {1/3 ; 2/3}.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and multiple answers provided by different educators. The question asks for the solutions of the equation \( f(-3x) = 0 \) given that the roots of \( f(x) = 0 \) are -1 and -2. Each educator provides a step-by-step solution, leading to the same answer: the solutions are \( x = \frac{1}{3} \) and \( x = \frac{2}{3} \).
  
  Label: question-and-answer