To accurately assess Jamal's and George's solutions to the quadratic equation \(x^2 + 4x - 5 = 7\), we first need to rewrite the equation in standard form by moving all terms to one side:
- Subtract 7 from both sides: \[ x^2 + 4x - 5 - 7 = 0 \] Simplifying, we get: \[ x^2 + 4x - 12 = 0 \]
Now we can factor the quadratic, set it equal to zero, and apply the Zero Product Property.
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We factor the equation \(x^2 + 4x - 12\): \[ (x + 6)(x - 2) = 0 \]
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Now apply the Zero Product Property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\):
- From \(x + 6 = 0\), we get \(x = -6\).
- From \(x - 2 = 0\), we get \(x = 2\).
The two solutions to the equation are \(x = -6\) and \(x = 2\).
Analyzing the work of Jamal and George:
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Correct student's solution: If one of the students correctly factored the equation \(x^2 + 4x - 12\) into \((x + 6)(x - 2) = 0\) and applied the Zero Product Property accurately to find \(x = -6\) and \(x = 2\), then that student’s solution is accurate.
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Incorrect student's solution: If the other student factored incorrectly (for instance, if they wrote the factors as \((x + 4)(x - 3) = 0\)) or set up the original equation improperly, their application of the Zero Product Property would yield incorrect solutions.
To determine who was correct, you can check both students’ factoring methods and their final solutions once they applied the Zero Product Property. Correct factorizations leading to \(x = -6\) and \(x = 2\) indicates an accurate solution, while incorrect factors or results would demonstrate an inaccurate approach.
In summary, the student with the correct factors and solutions demonstrates mastery of the Zero Product Property, while the other student’s incorrect factors or application reveals a misunderstanding or error in their solution process.
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