To determine where Hyung-eun may have made an error in solving the quadratic equation \( x^2 - 4x - 12 = 0 \) by grouping, we can follow the steps typically taken in this method.
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Rearrange the equation: The equation is already set to \( x^2 - 4x - 12 = 0 \).
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Factor the quadratic expression: To use grouping, the expression needs to be rewritten. The first step is to find two numbers that multiply to \(-12\) (the constant term) and add to \(-4\) (the coefficient of \(x\)). The factors of \(-12\) that work are \(2\) and \(-6\) since \(2 + (-6) = -4\).
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Rewrite the expression using these factors: \[ x^2 - 6x + 2x - 12 = 0 \] Grouping the terms: \[ (x^2 - 6x) + (2x - 12) = 0 \]
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Factor by grouping: Now factor out common terms from each group. \[ x(x - 6) + 2(x - 6) = 0 \]
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Combine the factored groups: \[ (x - 6)(x + 2) = 0 \]
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Solve for \( x \): Set each factor to zero: \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
If Hyung-eun follows through these steps, there seems to be no error up to this point. However, if you can provide her actual steps as she wrote them out, then I can pinpoint the exact moment in her process where she made an error.
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