Jamal

Question

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

To evaluate the accuracy of Jamal's and George’s solutions, we need to carefully analyze the steps of each student.

Jamal's Solution:

1. Initial equation: \( x^2 + 4x - 12 = 0 \)
2. Factored form: \( (x + 6)(x - 2) = 0 \)
3. Setting factors to zero:
   • \( x + 6 = 0 \) and \( x - 2 = 0 \)
4. Solutions:
   • From \( x + 6 = 0 \), we get \( x = -6 \)
   • From \( x - 2 = 0 \), we get \( x = 2 \)

George's Solution:

1. Initial equation: \( x^2 + 4x - 5 = 7 \)
2. Rearranging: First, to make it a standard quadratic equation, we would need to set it to 0: \( x^2 + 4x - 5 - 7 = 0 \) becomes \( x^2 + 4x - 12 = 0 \) (which is the same equation as Jamal’s).
3. Factored form: George states it as \( (x + 5)(x - 1) = 7 \), which is incorrect since the standard form should be equal to 0, not 7.
4. Setting factors to zero:
   • He sets \( x + 5 = 0 \) and \( x - 1 = 0 \), which is inconsistent with the previous correct factorization.
5. Solutions:
   • From \( x + 5 = 0 \), he wrongly concludes \( x = -5 \)
   • From \( x - 1 = 0 \), he incorrectly concludes \( x = 1 \)

Analysis:

• Jamal's Correct Approach: His initial equation was already set to 0, and his factors \( (x + 6)(x - 2) \) are valid for the equation \( x^2 + 4x - 12 = 0 \). Setting each factor to zero leads to correct solutions of \( x = -6 \) and \( x = 2 \).

• George's Incorrect Approach: George's error arises from incorrect factorizations and failing to first set the equation equal to 0. He does not have the correct resulting equation, which leads to invalid factorization and incorrect solutions.

Conclusion:

Jamal's solutions are accurate and correctly apply the Zero Product Property to solve the quadratic equation, while George's solutions are inaccurate because of his initial misunderstanding and mishandling of the equation.