To assess the accuracy of Jamal's and George's solutions to the quadratic equation \( x^2 + 4x - 5 = 7 \), we can analyze the steps they took in solving the equation.
Jamal's Solution:
- Set the equation to 0: \( x^2 + 4x - 12 = 0 \)
- Factored it as: \( (x + 6)(x - 2) = 0 \)
- Set each factor equal to 0: \( x + 6 = 0 \) and \( x - 2 = 0 \)
- Solved: \( x = -6 \) and \( x = 2 \)
George's Solution:
- Started with the equation: \( x^2 + 4x - 5 = 7 \)
- Incorrectly transformed it into: \( (x + 5)(x - 1) = 7 \)
- Set each factor equal to 0: \( x + 5 = 0 \) and \( x - 1 = 0 \)
- Solved: \( x = -5 \) and \( x = 1 \)
Analysis:
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Jamal's first step is inaccurate because he transforms \( x^2 + 4x - 5 = 7 \) incorrectly into \( x^2 + 4x - 12 = 0 \). The correct step should have been \( x^2 + 4x - 5 - 7 = 0 \) or \( x^2 + 4x - 12 = 0 \), which is why this equation seems incorrect.
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George's second step is also problematic due to incorrect factoring. The equation \( x^2 + 4x - 12 = 0 \) should factor to \( (x + 6)(x - 2) = 0 \) rather than what George wrote, which incorrectly implies \( (x + 5)(x - 1) = 7 \).
Conclusion: To determine who is correct, we can start with George’s correct equation of \( x^2 + 4x - 12 = 0 \) and notice that he made a calculation mistake during the factoring process. Jamal also made a mistake when he tried to solve his equation.
Final Determination: The correct solution should either be derived from confirming the correct factorization of \( x^2 + 4x - 12 = (x + 6)(x - 2) = 0 \), leading to the accurate solutions \( x = -6 \) and \( x = 2 \). However, Jamal and George both contain flaws—George with his algebraic transformation and Jamal with his factorization approach. The correct pathway should go through finding \( x^2 + 4x - 12 = 0 \) correctly and following its factorization to solve.
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