To determine which student's solution is incorrect, we need to evaluate their work against the original equation step by step.
The original equation is: \[ x^2 + 4x - 5 = 7 \]
Step 1: Set the equation to 0. To do this, we first subtract 7 from both sides: \[ x^2 + 4x - 5 - 7 = 0 \] This simplifies to: \[ x^2 + 4x - 12 = 0 \]
Step 2: Factor the quadratic equation. Now we need to factor \( x^2 + 4x - 12 \). We are looking for two numbers that multiply to -12 and add to 4. These numbers are 6 and -2, so the factored form is: \[ (x + 6)(x - 2) = 0 \]
Step 3: Set each factor equal to 0. We set each factor equal to zero:
- \( x + 6 = 0 \) leads to \( x = -6 \)
- \( x - 2 = 0 \) leads to \( x = 2 \)
Step 4: Solve each equation. From the above step, we find the solutions: \[ x = -6 \quad \text{and} \quad x = 2 \]
Now let's look at George's solution:
- George incorrectly left the equation in the form \( x^2 + 4x - 5 = 7 \) without setting it to zero.
- His factors \( (x + 5)(x - 1) = 7 \) are not correct for the equation \( x^2 + 4x - 12 = 0 \). The correct factors should lead to the left-hand side being equal to zero, not seven.
- When he sets each factor equal to zero, he solves \( (x + 5) = 0 \) which yields \( x = -5 \) and \( (x - 1) = 0 \) which yields \( x = 1 \), neither of which are correct.
Based on this evaluation, George is the incorrect student because he did not properly set the equation to zero before factoring it, resulting in incorrect factors and solutions. Jamal's method and final answers of \( x = -6 \) and \( x = 2 \) are accurate.
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