Step 1: Set equation =0

Question

. 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=0and 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)

Answer 1

To assess the accuracy of Jamal and George's solutions, we need to evaluate the correctness of their respective steps, especially when they wrote the quadratic equation in its factored form.

George's Solution:

1. Original Equation:
   \(x^2 + 4x - 5 = 7\)

2. Rearrance:
   Rearranging the equation gives us:
   \(x^2 + 4x - 5 - 7 = 0\)
   or
   \(x^2 + 4x - 12 = 0\)

3. Factoring:
   George provided the factorization as:
   \((x + 5)(x - 1) = 7\)
   However, this is not correct as we should factor the equation \(x^2 + 4x - 12 = 0\) into:
   \((x + 6)(x - 2) = 0\)

4. Setting Each Factor to Zero:
   He incorrectly set the factors derived from a wrong factorization.

Jamal's Solution:

1. Original Equation:
   \(x^2 + 4x - 5 = 7\)

2. Rearrance:
   Similarly, rearranging the equation correctly leads us to:
   \(x^2 + 4x - 5 - 7 = 0\)
   which simplifies to
   \(x^2 + 4x - 12 = 0\)

3. Factoring:
   Jamal's work includes an incorrect factorization as well, specifically:
   \((x + 6)(x - 2) = 0\)
   This factorization is indeed correct for \(x^2 + 4x - 12 = 0\).

4. Setting Each Factor to Zero:
   \(x + 6 = 0\) and \(x - 2 = 0\)

5. Solving Each Equation:
   He provides the correct roots:
   \(x = -6\) and \(x = 2\).

Conclusion:

• George's Work: The factorization he provided is incorrect, as it derives from a misapplication of the Zero Product Property. Thus, his roots \(x = -5\) and \(x = 1\) are not valid.

• Jamal's Work: Although he also started with the same modified equation, he correctly factored the quadratic \(x^2 + 4x - 12 = 0\) into \((x + 6)(x - 2) = 0\). This adheres to the Zero Product Property, leading to accurate solutions \(x = -6\) and \(x = 2\).

Therefore, Jamal's solution is accurate, while George's is inaccurate due to incorrect factorization and setting the wrong equation components.

Answer 2

shorten

Answer 3

To determine the accuracy of Jamal's and George's solutions:

George's Solution:

1. Equation Rearrangement:
   He correctly rewrote \(x^2 + 4x - 5 = 7\) as \(x^2 + 4x - 12 = 0\).

2. Incorrect Factoring:
   George stated the factorization as \((x + 5)(x - 1) = 0\), which is incorrect.

3. Incorrect Roots:
   His solutions \(x = -5\) and \(x = 1\) are invalid due to the wrong factorization.

Jamal's Solution:

1. Correct Rearrangement:
   He also rewrote the equation as \(x^2 + 4x - 12 = 0\).

2. Correct Factoring:
   Jamal correctly factored the equation as \((x + 6)(x - 2) = 0\).

3. Correct Roots:
   His solutions \(x = -6\) and \(x = 2\) are valid.

Conclusion:

Jamal's solution is accurate, while George's is inaccurate due to incorrect factorization and resulting roots.