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 determine the accuracy of Jamal's and George's solutions, let's review their respective steps in solving the equations.

1. Jamal's Equation:

   • Jamal started with the equation \(x^2 + 4x - 12 = 0\).
   • He factored it as \((x + 6)(x - 2) = 0\).
   • Setting each factor to zero gave him \(x + 6 = 0\) and \(x - 2 = 0\), leading to the solutions \(x = -6\) and \(x = 2\).

2. George's Equation:

   • George started with the equation \(x^2 + 4x - 5 = 7\).
   • He incorrectly set this equation up as \(x^2 + 4x - 5 - 7 = 0\), which simplifies to \(x^2 + 4x - 12 = 0\) (this is equivalent to Jamal's), but he misconfigured the subsequent steps.
   • In step 2, George attempted to factor the equation as \((x + 5)(x - 1) = 7\) which is incorrect; it should simply not include \(= 7\).
   • Then, he set each supposed factor to zero: \(x + 5 = 0\) and \(x - 1 = 0\), leading to incorrect solutions \(x = -5\) and \(x = 1\).

Explanation of Accuracy/Accuracy:

1. Correctness of Jamal's solution:

   • Jamal accurately identified the equation and successfully factored it correctly. He applied the Zero Product Property correctly by setting the factors equal to zero, yielding solutions that are derived from that factored form of the equation.

2. Errors in George's solution:

   • George made an error in his factorization stage. When he stated that \( (x + 5)(x - 1) = 7\), it violated the principles of the Zero Product Property, which applies only to equations set equal to zero (not equal to 7).
   • He also did not correctly manipulate the original equation to get it in standard form before attempting to factor.

Conclusion:

The key point is that Jamal came from the accurate factorization of the correct transformed equation into a product, allowing for legitimate solutions through applying the Zero Product Property correctly, whereas George's misinterpretation of the factorization step resulted in an incorrect equation and thus incorrect solutions.