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.

(2 points)

Answer 1

To analyze which student's solution is correct, we need to closely examine the setup and the steps involved in solving the quadratic equation \( x^2 + 4x - 5 = 7 \).

George's Solution:

1. Beginning with the equation:
   George starts with \( x^2 + 4x - 5 = 7 \).

2. Rearranging the equation:
   To apply the Zero Product Property correctly, one must first move all terms to one side to set the equation to zero. Thus, George should have rewritten the equation as: \[ x^2 + 4x - 5 - 7 = 0 \quad \Rightarrow \quad x^2 + 4x - 12 = 0 \]
   However, he incorrectly factored as \( (x+5)(x-1) = 7 \), which is incorrect because \( 7 \) should not be in the equation at this stage as the equation hasn't been set to zero.

3. Setting factors equal to zero:
   By setting \( (x+5)(x-1) = 7 \), George's factors are not properly derived to accommodate the Zero Product Property but rather set equal to a number, which is not correct in this context.

Jamal's Solution:

1. Beginning with the equation:
   Jamal starts with the same quadratic form but correctly sets up his equation to zero: \[ x^2 + 4x - 12 = 0 \] This was the correct form that George should have achieved.

2. Factoring:
   Jamal correctly factors the equation as \( (x + 6)(x - 2) = 0 \), which provides two potential solutions.

3. Using the Zero Product Property:
   From \( (x + 6)(x - 2) = 0 \), he sets each factor equal to zero:

   • \( x + 6 = 0 \Rightarrow x = -6 \)
   • \( x - 2 = 0 \Rightarrow x = 2 \)

Conclusion:

• Jamal's Solution is Accurate: He correctly set the quadratic equation to zero and factored it properly to find the roots.
• George's Solution is Inaccurate: He did not correctly set the equation to zero before attempting to factor, leading to incorrect factors and solutions. Thus, he utilized the Zero Product Property incorrectly since the setup for it was flawed.

In summary, the key to determining the accuracy lies in observing that Jamal correctly transformed the equation into standard form and factored it appropriately before applying the Zero Product Property. In contrast, George misapplied the Zero Product Property by failing to correctly set the equation to zero first.

Answer 2

make the conclusion sound like a 9th grade one

Answer 3

Conclusion:

Jamal's solution is accurate because he correctly set the equation \(x^2 + 4x - 5 = 7\) to zero by rewriting it as \(x^2 + 4x - 12 = 0\) and then factored it as \((x + 6)(x - 2) = 0\). This method allowed him to find the correct answers, \(x = -6\) and \(x = 2\), by using the Zero Product Property properly.

On the other hand, George's solution is inaccurate because he didn't set the equation to zero before trying to factor it. He ended up incorrectly stating the factors as \((x + 5)(x - 1) = 7\), which doesn't follow the right steps. This mistake meant he couldn't use the Zero Product Property correctly, resulting in wrong answers. So, Jamal did it right, while George made a mistake in his approach!