Use the tables to answer the question.

Question

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.

Answer 1

Bot GPT 4-o mini answered
2 hours ago

To determine which student provided the correct solution, let's first analyze each student's steps and identify any mistakes.

Jamal's Work:

Set the equation = 0: He correctly transformed the equation \(x^2 + 4x - 5 = 7\) into \(x^2 + 4x - 12 = 0\) by subtracting 7 from both sides.
Factoring: He factored the quadratic equation as \((x + 6)(x - 2) = 0\).
Set each factor to zero:
- \(x + 6 = 0 \Rightarrow x = -6\)
- \(x - 2 = 0 \Rightarrow x = 2\)

George's Work:

Original equation: He started with the equation \(x^2 + 4x - 5 = 7\), but he didn't correctly set the equation to zero. His step should have been \(x^2 + 4x - 12 = 0\), but he mistakenly wrote it as \((x + 5)(x - 1) = 7\), which is not correct because he left 7 on the right.
Factoring: The incorrect factorization led him to factor as \((x + 5)(x - 1) = 7\), which is incorrect.
Set each factor to zero:
- \(x + 5 = 0 \Rightarrow x = -5\)
- \(x - 1 = 0 \Rightarrow x = 1\)

Conclusion:

Jamal's Solution: His work accurately reflected the process of factoring the quadratic equation derived from setting it equal to zero. He correctly identified that the roots of the equation \(x^2 + 4x - 12 = 0\) were \(x = -6\) and \(x = 2\).
George's Solution: He made an error by not correctly setting the equation equal to zero and incorrectly factoring it. As a result, his solutions \(x = -5\) and \(x = 1\) do not come from a valid quadratic equation formed from the original.

In summary, Jamal's solution is correct because it follows the proper steps for applying the Zero Product Property on the correctly transformed quadratic equation. George's solution is incorrect because of a mistake made early in his process when he did not properly set the equation to zero.