To determine which student is incorrect, we need to analyze the steps taken by both Jamal and George in solving the quadratic equation \( x^2 + 4x - 5 = 7 \).
Step 1: Set the Equation to Zero
Initially, we need to move all terms to one side of the equation to set it equal to zero. Starting with \( x^2 + 4x - 5 = 7 \):
\[ x^2 + 4x - 5 - 7 = 0 \]
This simplifies to:
\[ x^2 + 4x - 12 = 0 \]
Both Jamal and George correctly set the equation to zero in Step 1.
Step 2: Factor the Quadratic Equation
Now let's look at the factoring of the quadratic \( x^2 + 4x - 12 \).
Jamal's factors: Jamal factored the equation as:
\[ (x + 6)(x - 2) = 0 \]
George's factors: George factored the equation as:
\[ (x + 5)(x - 1) = 7 \]
Here, George's factorization is incorrect for the equation \( x^2 + 4x - 12 = 0 \) because \( (x + 5)(x - 1) \) does not equal \( x^2 + 4x - 12 \).
Step 3: Setting Each Factor to Zero
Jamal set each factor correctly equal to 0:
\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
George's equations are based on his incorrect factorization:
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
Conclusion
Jamal is correct, and his solutions are \( x = -6 \) and \( x = 2\).
George is incorrect because his factorization \( (x + 5)(x - 1) \) does not yield \( x^2 + 4x - 12 = 0 \). The correct factors should lead to a factoring that satisfies the original quadratic equation correctly, and hence George's solutions of \( x = -5 \) and \( x = 1 \) are also incorrect.
Therefore, George is the student who is incorrect.
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