Use the tables to answer the question.

Question

Jamal
Step 1: Set equation =0
. x2+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. x2+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 x2+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 determine the accuracy of Jamal's and George's solutions, we need to analyze the steps they each took, particularly focusing on the equivalence of their equations and the application of the Zero Product Property.

Jamal's Solution:

1. Equation: \(x^2 + 4x - 12 = 0\)
   • Factoring: \( (x + 6)(x - 2) = 0 \)
   • Setting factors to zero:
     • \( x + 6 = 0 \) gives \( x = -6 \)
     • \( x - 2 = 0 \) gives \( x = 2 \)

George's Solution:

1. Equation: \(x^2 + 4x - 5 = 7\)

   • Rearranging the equation: This step is not completed in the table but should lead to \(x^2 + 4x - 12 = 0\).
   • Factoring**: \( (x + 5)(x - 1) = 7 \)
   • This expression is incorrect because George did not set the original equation to zero to apply the Zero Product Property correctly.

2. Setting factors to zero:

   • \( x + 5 = 0 \) gives \( x = -5 \)
   • \( x - 1 = 0 \) gives \( x = 1 \)

Analysis of the Two Solutions:

• Jamal's Solution: His steps begin correctly because he starts with the manipulated form of \(x^2 + 4x - 12 = 0\). This equation is considered properly prepared for applying the Zero Product Property. The factors he derived are also correct, leading to the solutions \(x = -6\) and \(x = 2\).

• George's Solution: George's formulation of the factors \( (x + 5)(x - 1) = 7 \) is fundamentally incorrect because he did not set the equation to zero first. To correctly apply the Zero Product Property, you must have an equal to zero state on one side. Therefore, his factoring is invalid in relation to the equation he's solving.

Conclusion:

Jamal's solution is accurate because he correctly transformed the original equation into the standard form required for factoring and subsequently used the Zero Product Property appropriately. George's approach is inaccurate as he did not correctly set the equation equal to zero before attempting to factor, leading to incorrect factorizations and solutions.