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.
To determine the accuracy of each student's solution, we can check if their final solutions satisfy the original equation.
For Jamal's solution:
The equation is x^2 + 4x - 12 = 0, and he factored it as (x + 6)(x - 2) = 0.
Setting each factor equal to 0, he obtained x + 6 = 0 and x - 2 = 0.
Solving each equation, he found x = -6 and x = 2.
To check his solution, we substitute these values back into the original equation:
For x = -6: (-6)^2 + 4(-6) - 12 = 0, which is true.
For x = 2: (2)^2 + 4(2) - 12 = 0, which is also true.
Since both values satisfy the original equation, we can conclude that Jamal's solution is accurate.
For George's solution:
The equation is x^2 + 4x - 5 = 7, and he factored it as (x + 5)(x - 1) = 7.
Setting each factor equal to 0, he obtained x + 5 = 0 and x - 1 = 0.
Solving each equation, he found x = -5 and x = 1.
To check his solution, we substitute these values back into the original equation:
For x = -5: (-5)^2 + 4(-5) - 5 = 10 - 20 - 5 = -15, which is not equal to 7.
For x = 1: (1)^2 + 4(1) - 5 = 1 + 4 - 5 = 0, which is not equal to 7.
Since both values do not satisfy the original equation, we can conclude that George's solution is inaccurate.
To determine the accuracy of the solutions, we need to evaluate if the solutions satisfy the original equation, x^2 + 4x - 5 = 7.
For Jamal:
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
x - 2 = 0
Step 4: Solve each equation
x = -6
x = 2
We can check if these solutions satisfy the original equation:
For x = -6:
(-6)^2 + 4(-6) - 5 = 36 - 24 - 5 = 7 (satisfied)
For x = 2:
(2)^2 + 4(2) - 5 = 4 + 8 - 5 = 7 (satisfied)
Both solutions satisfy the original equation. Therefore, Jamal's solution is accurate.
For 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
x - 1 = 0
Step 4: Solve each equation
x = -5
x = 1
We can check if these solutions satisfy the original equation:
For x = -5:
(-5)^2 + 4(-5) - 5 = 25 - 20 - 5 = 0 (not satisfied)
For x = 1:
(1)^2 + 4(1) - 5 = 1 + 4 - 5 = 0 (not satisfied)
Neither solution satisfies the original equation. Therefore, George's solution is inaccurate.
To determine the accuracy of the solutions provided by Jamal and George, we need to analyze their steps and compare them with the correct method of solving quadratic equations.
Let's first look at Jamal's solution:
Step 1: Setting the equation equal to 0 is the correct first step in solving a quadratic equation.
Step 2: Writing the quadratic equation as the product of two factors (x+6)(x−2)=0 is also correct. This is known as factoring the quadratic equation.
Step 3: Setting each factor equal to 0 (x+6=0 and x−2=0) is another correct step. This allows us to solve for the values of x that make each factor equal to 0.
Step 4: Solving each equation (x=−6 and x=2) is the final step, and the values obtained are indeed the solutions to the equation x^2+4x−12=0.
Now let's examine George's solution:
Step 1: Starting with the equation x^2+4x−5=7 is correct.
Step 2: Writing the quadratic equation as (x+5)(x−1)=7 is incorrect. George made a mistake in factoring the equation, as the factors should be (x+5)(x−1)=0, not 7. This mistake affects the accuracy of the further steps.
Step 3: Setting each factor equal to 0 (x+5=0 and x−1=0) is correct, but it should be done assuming the product is equal to zero.
Step 4: Solving each equation (x=−5 and x=1) is correct, but these values are not solutions to the original equation x^2+4x−5=7. They are solutions to the incorrect equation (x+5)(x−1)=0.
Based on these analyses, we can conclude that Jamal's solution is accurate because it follows the correct steps to solve a quadratic equation. On the other hand, George's solution is inaccurate because he made an error in factoring the equation, which led to incorrect solutions.