To determine which student's solution is accurate and which is inaccurate, we should analyze their work step by step, particularly focusing on how they set up their equations, their factorization, and the application of the Zero Product Property.
Jamal's Solution:
- Set equation = 0: Jamal correctly starts with the equation \(x^2 + 4x - 12 = 0\).
- Writing it as a product of two factors: He factors it as \((x + 6)(x - 2) = 0\).
- This factorization is accurate because \((x + 6)(x - 2)\) expands to \(x^2 + 4x - 12\).
- Set each factor to zero: Setting \(x + 6 = 0\) and \(x - 2 = 0\) is correct.
- Solving each equation: He solves correctly, obtaining \(x = -6\) and \(x = 2\).
George's Solution:
- Begin with the equation: George starts with \(x^2 + 4x - 5 = 7\).
- Writing it as a product of two factors: Instead of setting the equation to zero, he does not subtract 7 from both sides. The equation should first be written as \(x^2 + 4x - 12 = 0\) (after correcting for the equality).
- George incorrectly factors it as \((x + 5)(x - 1) = 7\). This factorization is incorrect because this does not correctly represent the left-hand side.
- Setting each factor to zero: Setting \(x + 5 = 0\) and \(x - 1 = 0\) is based on an incorrect factorization.
- Solving each equation: He gets \(x = -5\) and \(x = 1\) based on incorrect factors.
Conclusion:
- Jamal's solution is accurate because he correctly handled the initial quadratic equation, factored it correctly, and applied the Zero Product Property accurately.
- George's solution is inaccurate due to failing to correctly set the quadratic equation to zero before factoring. His factorization was based on the wrong expression, leading to incorrect solutions.
Thus, we conclude that Jamal's method and answers are correct, while George's approach reveals a misunderstanding in handling quadratic equations and the Zero Product Property.
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