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Use the tables to answer the 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.

1 answer

To determine the accuracy of Jamal's and George's solutions, let's analyze their steps in relation to the quadratic equation they worked on.

George's Steps on the Equation \( x^2 + 4x - 5 = 7 \)

  1. Setting the equation to zero: George starts with the equation \( x^2 + 4x - 5 = 7 \). To apply the Zero Product Property, he should first set the equation to zero by subtracting 7 from both sides: \[ x^2 + 4x - 5 - 7 = 0 \quad \Rightarrow \quad x^2 + 4x - 12 = 0. \] However, he did not perform this step correctly.

  2. Factoring: Next, George factors the expression incorrectly. Instead of factoring \( x^2 + 4x - 12 \), which is what he should have, he indicated that he factored \( x^2 + 4x - 5 \) as \( (x+5)(x-1) = 7 \), which is incorrect.

  3. Setting the factors equal to 0: Since the previous step was incorrect, this also becomes invalid.

  4. Solving incorrectly: Finally, the roots he obtained from incorrect factors (\( x = -5 \) and \( x = 1 \)) are not valid solutions of the original equation after it is set to zero.

Jamal's Steps on the Equation \( x^2 + 4x - 12 = 0 \)

  1. Setting the equation to zero: Jamal successfully sets the equation \( x^2 + 4x - 12 = 0 \) correctly after recognizing it needed adjustment from \( x^2 + 4x - 5 = 7 \).

  2. Factoring: He correctly factored \( x^2 + 4x - 12 \) as \( (x+6)(x-2) = 0 \).

  3. Setting the factors equal to 0: He accurately sets each factor to zero; \( x + 6 = 0 \) and \( x - 2 = 0 \).

  4. Solving correctly: He then solves them to find that \( x = -6 \) and \( x = 2 \).

Conclusion

  • Correct Solution: Jamal's work follows the correct procedure, beginning with the adjustment to get the equation set to zero and correctly factoring the resulting expression. His solutions are valid within the context of the quadratic equation.

  • Incorrect Solution: George's work reflected errors from the beginning—he did not set the equation to zero and incorrectly factored an expression unrelated to the original problem. Because he didn’t apply the Zero Product Property to the correct equation, his solutions are invalid.

Jamal's thoroughness and accuracy in applying the right steps demonstrate that his solution is valid while George’s incorrect assumptions led to an erroneous solution.