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Students are presented with an AI generated solution to rerrange the quadratic equation where the AI has made errors, they are asked to identify on which line the first error occurs, then rewrite the solution correctly. No variables but this is version 4 of 5 versions of the question.

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The first error occurs at line 3 where we should add \\(b\\) to each side of the equation, not subtract it.

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The correct solution starting at line 3 is:

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  1. Next, we'll isolate the square root term by adding\\(b\\) to both sides: \\(2ax + b = \\sqrt{b^2 - 4ac}\\)
    2. \n
  2. Now we'll square both sides to eliminate the square root: \\((2ax + b)^2 = b^2 - 4ac\\)
    3. \n
  3. Expanding the left side of the equation gives: \\(4a^2x^2 + 4abx + b^2 = b^2 - 4ac\\)
    4. \n
  4. Simplifying further by canceling out the \\(b^2\\) terms: \\(4a^2x^2 + 4abx = -4ac\\)
    5. \n
  5. Factoring out \\(4a: 4a (ax^2 + bx) = -4ac\\) (Note that there is a transcription error here in the AI solution as well)
    6. \n
  6. Dividing both sides by \\(-4a\\) and switching the sides: \\(c = -ax^2-bx\\) (The AI has also made an error with this division)
    7. \n
  7. Therefore, \\(c\\) is equal to \\(-ax^2-bx.\\)
    8. \n
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I asked an AI chat program to rearrange \\(\\displaystyle{x=\\frac{-b+\\sqrt{b^2-4ac}}{2a}}\\) to make \\(c\\) the subject. The response was:

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  1. Starting with the formula: \\(\\displaystyle{x = \\frac{-b + \\sqrt{b^2 - 4ac}}{ 2a}}\\)
    2. \n
  2. First, we'll multiply both sides by \\(2a\\) to eliminate the denominator: \\(2ax = -b + \\sqrt{b^2 - 4ac}\\)
    3. \n
  3. Next, we'll isolate the square root term by subtracting \\(b\\) from both sides: \\(2ax - b = \\sqrt{b^2 - 4ac}\\)
    4. \n
  4. Now we'll square both sides to eliminate the square root: \\((2ax - b)^2 = b^2 - 4ac\\)
    5. \n
  5. Expanding the left side of the equation gives: \\(4a^2x^2 - 4abx + b^2 = b^2 - 4ac\\)
    6. \n
  6. Simplifying further by canceling out the \\(b^2\\) terms: \\(4a^2x^2 - 4abx = -4ac\\)
    7. \n
  7. Factoring out \\(4a: 4a (ax - b) = -4c\\)
    8. \n
  8. Dividing both sides by \\(-4\\) and switching the sides: \\(c = (ax - b) / 4a\\)
    9. \n
  9. Therefore, \\(c\\) is equal to \\((ax - b) / 4a.\\)
    10. \n
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On which line does the first error occur? In your handwritten notes write out the solution to this problem correctly.

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