Use the tables to answer the question.

Question

Jamal
Step 1: Set equation =0
. x^2+4x−12=0
Step 2: Write quadratic equation as the product of two factors. (x+6)(x−2)=0
Step 3: Set each factor equal to 0. x+6=0
and x−2=0
Step 4: Solve each equation. x=−6
and x=2
George
Step 1: Begin with the equation. x^2+4x−5=7
Step 2: Write quadratic equation as the product of two factors. (x+5)(x−1)=7
Step 3: Set each factor equal to 0. x+5=0
and x−1=0
Step 4: Solve each equation. x=−5
and x=1
When trying to solve the quadratic equation x^2+4x−5=7
, Jamal and George each provided their work in the tables. Each said they used the Zero Product Property to solve the equation after step 2. Explain how you know the correct student’s solution is accurate and how you know the incorrect student’s solution is inaccurate.

Answer 1

To evaluate the correctness of Jamal's and George's solutions, we must analyze their approaches to the quadratic equation \( x^2 + 4x - 5 = 7 \) and how they applied the Zero Product Property.

1. Rearranging the equation: Both students should have started by rearranging the equation: \[ x^2 + 4x - 5 - 7 = 0 \implies x^2 + 4x - 12 = 0. \] This correction is crucial for both Jamal and George because they are meant to solve the corrected form of the equation.

2. Identifying the factors:

   • Jamal's Description: He correctly factored the equation as \[ (x + 6)(x - 2) = 0. \] Using the Zero Product Property, he set each factor equal to zero resulting in \( x + 6 = 0 \) (thus \( x = -6 \)) and \( x - 2 = 0 \) (thus \( x = 2 \)).

   However, his use of factors is based on the incorrect equation \( x^2 + 4x - 12 = 0 \) which is different from what George and the initial problem stated.

   • George's Description: George attempted to factor the original equation \( x^2 + 4x - 5 - 7 \) as \[ (x + 5)(x - 1) = 7. \] This step is incorrect because he should have set the equation to equal zero first (as Jamal’s step implied).

3. Conclusion:

   • Correct Student's Solution: In reality, the correct factoring of the equation \( x^2 + 4x - 12 = 0 \) should yield \[ (x + 6)(x - 2) = 0. \] Thus, Jamal should have calculated: \[ x = -6 \quad \text{and} \quad x = 2. \] This means his factors are accurate for the equation \( x^2 + 4x - 12 = 0 \), but the logic flow was based on an incorrect start from the original equation.

   • Incorrect Student's Solution: George's factorization was incorrect because he split it incorrectly based on the wrong version of the equation.

Therefore, neither solution initially portrays the right combinations correctly but Jamal ends up with the right roots for his derived equation. The right approach would have entailed both students starting correctly from the rearrangement to yield \( x^2 + 4x - 12 = 0 \) first, but only Jamal correctly factored the quadratic derived but neglecting the alignment with the original problem which made it inaccurate in contention with the true original equation starting point.