To determine where Hyung-eun first makes an error in solving the quadratic equation \( x^2 - 4x - 12 = 0 \) by grouping, let's analyze the steps of grouping we might expect in solving the equation.
- Rearranging the equation: The equation is \( x^2 - 4x - 12 = 0 \).
- Factoring the quadratic: The goal is to find two numbers that multiply to \( -12 \) (the constant term) and add to \( -4 \) (the coefficient of \( x \)). Those numbers are \( -6 \) and \( 2 \).
- Writing the quadratic in factored form: It becomes \( (x - 6)(x + 2) = 0 \).
- Finding the solutions: Set each factor equal to zero:
- \( x - 6 = 0 \) gives \( x = 6 \).
- \( x + 2 = 0 \) gives \( x = -2 \).
If Hyung-eun's steps include any of these incorrectly, that would indicate her error.
Common mistakes include:
- Miscalculating the pairs of numbers that sum and multiply correctly.
- Incorrectly grouping or distributing terms.
If we assume Hyung-eun's work involves incorrect factorization or missing a sign while performing these steps, the first error would typically happen at the factoring step where she identifies or combines terms incorrectly.
If you have specific steps of her work shown, I can help identify the exact error based on that.
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