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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 2 of 5 versions of the question.

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The error occurs at line 2 of the solution where \\(b\\) should be added rather than subtracted from each side.

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The corrected solution from line 2 is:

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  1. Next, we can isolate the square root term by adding b to both sides:\\(2ax + b = \\sqrt{b^2 - 4ac}\\)
    2. \n
  2. Then, we can square both sides to eliminate the square root:\\((2ax + b)^2 = b^2 - 4ac\\)
    3. \n
  3. Expanding the left-hand side:\\(4a^2x^2 + 4abx + b^2 = b^2 - 4ac\\)
    4. \n
  4. Simplifying:\\(4a^2x^2 + 4abx = -4ac\\)
    5. \n
  5. Finally, we can isolate \\(c\\) by dividing both sides by \\(-4a: c = -ax^2-bx\\) (note there is a second error here in the AI answer)
    6. \n
  6. Therefore, the formula rearranged to make \\(c\\) the subject is:\\(c = c = -ax^2-bx\\)
    7. \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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Starting with the formula: \\(\\displaystyle{x = \\frac{-b + \\sqrt{b^2 - 4ac}}{ 2a}}\\)

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We want to rearrange it to make \\(c\\) the subject.

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  1. We can start by multiplying both sides by \\(2a:  2ax = -b + \\sqrt{b^2 - 4ac}\\) 
    2. \n
  2. Next, we can isolate the square root term by subtracting b from both sides:\\(2ax - b = \\sqrt{b^2 - 4ac}\\)
    3. \n
  3. Then, we can square both sides to eliminate the square root:\\((2ax - b)^2 = b^2 - 4ac\\)
    4. \n
  4. Expanding the left-hand side:\\(4a^2x^2 - 4abx + b^2 = b^2 - 4ac\\)
    5. \n
  5. Simplifying:\\(4a^2x^2 - 4abx = -4ac\\)
    6. \n
  6. Finally, we can isolate \\(c\\) by dividing both sides by \\(-4a: c = \\frac{4a^2x^2 - 4abx}{-4a}\\)
    7. \n
  7. Simplifying:\\(c = -x^2 + bx / a\\)
    8. \n
  8. Therefore, the formula rearranged to make \\(c\\) the subject is:\\(c = -x^2 + bx / a\\)
    9. \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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