To identify the error in Hyung-eun's steps when solving the quadratic equation \(x^2 - 4x - 12 = 0\) by grouping, we can analyze each step:
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Original Equation: \(x^2 - 4x - 12 = 0\)
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Step 1: \(x^2 + (-6x) + 2x - 12 = 0\)
- This step factors the middle term incorrectly. The correct grouping should lead to two numbers that multiply to \(-12\) and add up to \(-4\). The numbers are \(-6\) and \(2\), but they should combine to be \(x^2 - 6x + 2x - 12\).
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Step 2: \((x^2 + (-6x)) + (2x - 12) = 0\)
- This step is simply regrouping the previous expression, so it is still fine.
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Step 3: \(x(x + (-6)) + 2(x - 6) = 0\)
- This also seems to be a correct grouping step.
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Step 4: \((x + 6)(x - 2) = 0\)
- This step is where the significant error occurs because the product should not match the factors derived from the earlier steps. The correct factorization should have been \((x - 6)(x + 2)\).
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Step 5: Solving for the roots based on an incorrect factorization leads to incorrect solutions.
Given this analysis, the first error occurs in Step 1, where she incorrectly breaks down the middle term into \((-6x)\) and \(2x\). Therefore, the correct answer is:
Step 1 - She incorrectly grouped the terms.
1 answer