To determine where Hyung-eun first makes an error in solving the quadratic equation \( x^2 - 4x - 12 = 0 \) using grouping, let's analyze each step provided:
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Step 1: \( x^2 + (-6x) + 2x - 12 = 0 \)
- This step involves rewriting \(-4x\) as \((-6x + 2x)\). This is correct since \(-6 + 2 = -4\).
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Step 2: \( (x^2 + (-6x)) + (2x - 12) = 0 \)
- This groups the first two terms and the last two correctly. So far, this step is also correct.
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Step 3: \( x(x + (-6)) + 2(x - 6) = 0 \)
- In this grouping, \( x^2 + (-6x) \) is factored correctly as \( x(x - 6) \), but \( 2x - 12 \) should be factored as \( 2(x - 6) \). Hyung-eun has still followed the steps correctly to this point.
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Step 4: \( (x + 6)(x - 2) = 0 \)
- This step is where the first error occurs. The correct factorization from Step 3 should yield: \[ x(x - 6) + 2(x - 6) = (x + 2)(x - 6) = 0 \] Instead, Hyung-eun wrote \((x + 6)(x - 2)\), which is incorrect.
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Step 5: \( x + 6 = 0 \text{ or } x - 2 = 0 \)
- This step follows incorrectly from Step 4's factorization error.
In conclusion, the first error occurs in Step 4, where the factors were misidentified as \((x + 6)(x - 2)\) instead of the correct factors \( (x + 2)(x - 6) \).