Use the tables to answer the 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.
From the given information, we know that Jamal and George both used the Zero Product Property to solve their respective quadratic equations in step 3. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero.
For Jamal's solution, the quadratic equation x^2 + 4x - 12 = 0 was factored into (x + 6)(x - 2) = 0. Setting each factor equal to zero gives x + 6 = 0 and x - 2 = 0. Solving these equations gives x = -6 and x = 2. To check if Jamal's solution is accurate, we can substitute these values back into the original equation x^2 + 4x - 12 = 0 and see if the equation holds true. If the equation holds true for both values, then Jamal's solution is accurate.
For George's solution, the quadratic equation x^2 + 4x - 5 = 7 was factored into (x + 5)(x - 1) = 7. Setting each factor equal to zero gives x + 5 = 0 and x - 1 = 0. Solving these equations gives x = -5 and x = 1. However, George made a mistake by adding 7 to both sides of the equation instead of subtracting it. To check if George's solution is accurate, we can substitute his values back into the original equation. If the equation does not hold true for both values, then George's solution is inaccurate.
Therefore, we can conclude that Jamal's solution is accurate, as it satisfies the original equation, while George's solution is inaccurate, as it does not satisfy the original equation.
To determine if the correct student's solution is accurate, we can substitute the values of x back into the original equation and check if it holds true.
For Jamal's solution:
Substituting x = -6 into the original equation x^2 + 4x - 5 = 7, we get:
(-6)^2 + 4(-6) - 5 = 7
36 - 24 - 5 = 7
7 = 7
Substituting x = 2 into the original equation x^2 + 4x - 5 = 7, we get:
(2)^2 + 4(2) - 5 = 7
4 + 8 - 5 = 7
7 = 7
Since both substitutions result in the equation holding true, we can conclude that Jamal's solution is accurate.
For George's solution:
Substituting x = -5 into the original equation x^2 + 4x - 5 = 7, we get:
(-5)^2 + 4(-5) - 5 = 7
25 - 20 - 5 = 7
0 = 7
Substituting x = 1 into the original equation x^2 + 4x - 5 = 7, we get:
(1)^2 + 4(1) - 5 = 7
1 + 4 - 5 = 7
0 = 7
Both of George's substitutions result in the equation not holding true. Therefore, we can conclude that George's solution is inaccurate.
To determine the accuracy of Jamal and George's solutions, we need to analyze their steps and evaluate if they followed the correct process.
Let's start with Jamal's solution:
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
Jamal accurately factored the quadratic equation into the product of two binomials, and correctly set each factor equal to zero before solving. Thus, we can conclude that Jamal's solution is accurate.
Now let's examine George's solution:
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
Here is the problem with George's solution: at step 2, he incorrectly multiplied both factors by 7 instead of setting the equation equal to 7. Consequently, his factored equation is inaccurate. As a result, his obtained solutions of x = -5 and x = 1 are incorrect.
In summary, we can establish the accuracy of Jamal's solution by confirming that his steps are correct and yield the correct solutions. Conversely, we can identify the inaccuracy of George's solution by examining his mistake in step 2, resulting in an incorrect factored equation and solutions.