To determine which student’s solution is accurate and which is not, we first need to correctly solve the quadratic equation \(x^2 + 4x - 5 = 7\).
Step 1: Rearrange the equation
We start by moving all terms to one side: \[ x^2 + 4x - 5 - 7 = 0 \] This simplifies to: \[ x^2 + 4x - 12 = 0 \]
Step 2: Factor the quadratic
Next, we need to factor the quadratic equation \(x^2 + 4x - 12\). We want two numbers that multiply to \(-12\) (the constant term) and add to \(4\) (the coefficient of the linear term). These numbers are \(6\) and \(-2\).
Thus, we can rewrite the equation as: \[ (x + 6)(x - 2) = 0 \]
Step 3: Apply the Zero Product Property
According to the Zero Product Property, we set each factor equal to zero: \[ x + 6 = 0 \quad \text{or} \quad x - 2 = 0 \] This gives us the solutions: \[ x = -6 \quad \text{and} \quad x = 2 \]
Comparison of Jamal and George's Work:
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Jamal's Solution:
- He correctly factored the equation to \((x + 6)(x - 2) = 0\), which is accurate based on our calculations. His solutions \(x = -6\) and \(x = 2\) are both correct.
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George's Solution:
- He incorrectly factored the equation as \((x + 5)(x - 1) = 0\). This factoring is not accurate for the equation \(x^2 + 4x - 12 = 0\). Because he factored it incorrectly, his solutions \(x = -5\) and \(x = 1\) are also incorrect.
Conclusion:
Jamal's solution is accurate because he has factored the quadratic correctly and derived the correct solutions. George's solution is inaccurate because he has factored the expression incorrectly, leading to wrong answers.