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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 rewrite the solution correctly. No variables but this is version 5 of 5 versions of the question. This version uses a much more mangled AI generated solution that the other 4 versions and does not ask for the line with the first error, just for the student to rewrite the solution correctly.

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To rearrange the quadratic formula \\(\\displaystyle{x=\\frac{-b+\\sqrt{b^2-4ac}}{2a}}\\) to make \\(c\\) the subject, follow these steps:

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  1. Start with the equation:\\(\\displaystyle{x=\\frac{-b+\\sqrt{b^2-4ac}}{2a}}\\)
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  2. Multiply both sides by \\(2a\\): \\(2ax = -b + \\sqrt{b^2 - 4ac}\\)
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  3. Add \\(b\\) to both sides: \\(2ax + b = \\sqrt{b^2 - 4ac}\\)
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  4. Square both sides: \\((2ax + b)^2 = b^2 - 4ac\\)
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  5. Expand the left sides: \\(4a^2x^2 + 4abx + b^2 = b^2 - 4ac\\)
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  6. Subtract \\(b^2\\) from both sides:\\(4a^2x^2 + 4abx = -4ac\\)
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  7. Divide both sides by \\(-4a\\) : \\(-ax^2 - bx = c\\)
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  8. Now, \\(c\\) is the subject of the equation: \\(c = -ax^2 - bx\\)
    9. \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 isolate the variable \\(c\\) on one side of the equation. To do this, we will follow a series of steps:

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Multiply both sides of the equation by \\(2a: 2a  x = -b + \\sqrt{b^2 - 4ac}\\)

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Subtract \\(\\sqrt{b^2 - 4ac}\\) from both sides of the equation: \\(2a  x - \\sqrt{b^2 - 4ac} = -b\\)

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Add \\(b\\) to both sides of the equation: \\(2ax - \\sqrt{b^2 - 4ac} + b = 0\\)

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Add \\(\\sqrt{b^2 - 4ac} - b\\) to both sides of the equation: \\(2ax + \\sqrt{b^2 - 4ac} - b = \\sqrt{b^2 - 4ac} - b\\)

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Simplify the right-hand side: \\(2ax + \\sqrt{b^2 - 4ac} - b = -4ac / (\\sqrt{b^2 - 4ac}) + b)\\)

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Divide both sides of the equation by \\(-4a: c = (\\sqrt{b^2 - 4ac} - b) / 2a - x / 2a\\)

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Therefore, the equation rearranged to make \\(c\\) the subject is: \\(c = (\\sqrt{b^2 - 4ac} - b) / 2a - x / 2a.\\)

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On your handwritten notes comment on this solution and correct any errors that have been made.

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