Use the tables to answer the question.
Jamal
Step 1: Set equation = 0.
x? + 4х - 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? + 4x - 5 = 7
Step 2: Write quadratic equation as the product of two factors.
（2+5） （2=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? + 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.
To verify the accuracy of the solutions, we can substitute the values of x obtained by each student back into the original equation and check if both sides of the equation are equal.
For Jamal's solution, x = -6 and x = 2.
Substituting x = -6 into the equation x² + 4x -12 = 0:
(-6)² + 4(-6) -12 = 36 -24 - 12 = 0
Since the left-hand side is equal to zero, Jamal's solution is accurate for the given equation.
Substituting x = 2 into the equation x² + 4x -12 = 0:
(2)² + 4(2) - 12 = 4 + 8 - 12 = 0
Again, the left-hand side is equal to zero, confirming that Jamal's solution is accurate for the given equation.
For George's solution, x = -5 and x = 1.
Substituting x = -5 into the equation x² + 4x -5 = 7:
(-5)² + 4(-5) -5 = 25 -20 - 5 = 0
The left-hand side is equal to zero, indicating that George's solution is accurate for the given equation.
Substituting x = 1 into the equation x² + 4x -5 = 7:
(1)² + 4(1) - 5 = 1 + 4 - 5 = 0
Once again, the left-hand side is equal to zero, confirming that George's solution is accurate for the given equation.
From the above analysis, it is clear that both solutions provided by Jamal and George are accurate for their respective equations.
Based on the provided tables, we can evaluate the solutions given by Jamal and George for the quadratic equation x² + 4x - 5 = 7.
For Jamal's solution:
Step 1: The equation is correctly set to 0, x² + 4x - 12 = 0.
Step 2: The quadratic equation is factored correctly, (x + 6)(x - 2) = 0.
Step 3: The factors are set equal to 0, x + 6 = 0 and x - 2 = 0.
Step 4: The solutions are found correctly, x = -6 and x = 2.
For George's solution:
Step 1: The equation is correctly represented, x² + 4x - 5 = 7.
Step 2: The quadratic equation's factoring, however, is incorrect as "(2+5) (2=1) = 7" is not a valid factorization.
Step 3: The factors are incorrectly set equal to 0, x + 5 = 0 and x - 1 = 0.
Step 4: The solutions are found as x = -5 and x = 1, which are incorrect due to the inaccurate factorization.
From the comparison, we can see that Jamal's solution is accurate because his factoring and subsequent solving steps are correct and consistent with the original equation. On the other hand, George's solution is inaccurate because his factoring step is incorrect, leading to incorrect solutions.
To determine the accuracy of the solutions, let's evaluate the steps taken by each student:
Jamal:
Step 1: Jamal correctly sets the equation x? + 4x - 12 = 0.
Step 2: Jamal factors the quadratic equation as (x + 6)(x - 2) = 0. This step is also correct.
Step 3: Jamal sets each factor equal to 0, which is x + 6 = 0 and x - 2 = 0. These steps are accurate.
Step 4: Jamal solves each equation correctly, yielding x = -6 and x = 2.
George:
Step 1: George begins with the equation x? + 4x - 5 = 7. So far, this step is correct.
Step 2: However, George writes an incorrect factorization of the equation as (2 + 5)(2 = 1) = 7. The factorization should be (x + 5)(x - 1) = 0.
Step 3: George sets each factor equal to 0, which is x + 5 = 0 and x - 1 = 0. These steps are accurate.
Step 4: George correctly solves each equation, resulting in x = -5 and x = 1.
Based on the analysis, Jamal's solution is accurate because he correctly set up the quadratic equation, accurately factored it, and solved the resulting equations correctly. Therefore, his solutions, x = -6 and x = 2, are accurate.
On the other hand, George's solution is inaccurate due to the incorrect factorization in step 2. Consequently, his solutions, x = -5 and x = 1, cannot be considered as accurate.
Overall, Jamal's solution is correct, while George's solution contains an error in step 2.