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Use the tables to answer the 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. (2 points)

1 answer

To determine the accuracy of Jamal's and George's solutions, we can analyze their steps and identify the errors made in the process.

Jamal's Solution:

  1. Set equation = 0: Jamal starts with the equation \(x^2 + 4x - 12 = 0\). This is correct, as it appears that he is solving a different quadratic equation (not the original one given), but we will analyze his steps as they are.
  2. Factoring: He factored \(x^2 + 4x - 12\) correctly as \((x + 6)(x - 2) = 0\).
  3. Setting Factors to Zero: He correctly set each factor to zero:
    • \(x + 6 = 0 \Rightarrow x = -6\)
    • \(x - 2 = 0 \Rightarrow x = 2\)

Although Jamal's final values for \(x\) are correct for the quadratic he solved, he didn't start with the correct equation \(x^2 + 4x - 5 = 7\).

George's Solution:

  1. Begin with the equation: George correctly starts with \(x^2 + 4x - 5 = 7\).
  2. Rearranging the equation: To use the Zero Product Property, he should have first rewritten the equation in standard form. Instead, he incorrectly states \((x + 5)(x - 1) = 7\) as the factored form, which is inaccurate. The correct factored form after moving 7 to the left side would be \(x^2 + 4x - 12 = 0\), but he stated an incorrect factorization based on the rearrangement.
  3. Setting Factors Equal to Zero: George incorrectly sets \( (x + 5)(x - 1) = 7 \) which is not valid. The correct application of the Zero Product Property would apply only if the equation were set up as \( (x + 5)(x - 1) = 0 \).

In conclusion:

  • George's solution is inaccurate because he improperly factored the equation and set it equal to 7 instead of 0.
  • Jamal's solution is accurate within the context of solving \(x^2 + 4x - 12 = 0\), but it is not the solution for the equation he claims to solve, as it is unrelated to the original equation given.

Thus, George's work is incorrect due to the wrong factoring and handling of the equation, while Jamal's factoring was correctly performed for a different equation. Therefore, Jamal's process led to accurate solutions, but for the wrong problem.