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 evaluate the correctness of Jamal's and George's solutions, we need to review their steps closely and analyze their expressions and results at each step.

Jamal's Solution:

1. Step 1: Jamal correctly set the equation \(x^2 + 4x - 12 = 0\) by rearranging the equation \(x^2 + 4x - 5 = 7\) to get \(x^2 + 4x - 12 = 0\).
2. Step 2: He factored the equation as \((x + 6)(x - 2) = 0\). To check: \[ (x + 6)(x - 2) = x^2 - 2x + 6x - 12 = x^2 + 4x - 12 \] This factorization is correct.
3. Step 3: Jamal sets each factor equal to zero: \(x + 6 = 0\) and \(x - 2 = 0\).
4. Step 4: He correctly solved these equations, giving solutions \(x = -6\) and \(x = 2\).

George's Solution:

1. Step 1: George started with the same original equation \(x^2 + 4x - 5 = 7\), but incorrectly rearranged it to \(x^2 + 4x - 5 = 7\) to obtain \(x^2 + 4x - 5 - 7 = 0\) or \(x^2 + 4x - 12 = 0\).

2. Step 2: George's factorization was \((x + 5)(x - 1) = 7\). This is not a correct factorization of \(x^2 + 4x - 12\). The correct factors of \(x^2 + 4x - 12\) are \((x + 6)(x - 2)\).

   • To check George's factorization: \[ (x + 5)(x - 1) = x^2 - x + 5x - 5 = x^2 + 4x - 5 \] This was supposed to equal 7, which it does not; in fact, it would have been the expression that George was supposed to match to zero after moving 7 over.

3. Step 3: Setting each factor equal to zero is irrelevant here because the factorization itself was incorrect.

4. Step 4: Solving these incorrect equations \(x + 5 = 0\) and \(x - 1 = 0\) gives incorrect solutions \(x = -5\) and \(x = 1\).

Conclusion:

• Accuracy of Jamal's Solution: Jamal correctly manipulated the original equation and used the Zero Product Property after properly factoring the rearranged quadratic equation. Thus, his solutions \(x = -6\) and \(x = 2\) are correct.

• Inaccuracy of George's Solution: Although George started from the correct original equation, he mismanaged the re-arrangement and incorrectly factored the equation, leading to erroneous factors set equal to zero. His solutions \(x = -5\) and \(x = 1\) were based on this incorrect factorization and are therefore incorrect.

This illustrates that both students approached the problem with some mathematical principles, but George's incorrect understanding of factorization led him to the wrong results. Hence, Jamal’s solution is accurate, while George’s is not.

Answer 2

make it simple like a kid wrote it with at least 3-4 sentences

Answer 3

Jamal did the math right. He started with the equation \(x^2 + 4x - 5 = 7\) and changed it to \(x^2 + 4x - 12 = 0\). Then he factored it correctly into \((x + 6)(x - 2) = 0\) and found the answers \(x = -6\) and \(x = 2\).

George also started with the same equation, but he made a mistake when he tried to factor it. He wrote it as \((x + 5)(x - 1) = 7\), which is wrong, so his answers \(x = -5\) and \(x = 1\) are not correct. That's how we know Jamal's solution is accurate, but George's is not.