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 which student's solution is accurate and which is inaccurate, we need to compare their solutions and reasoning.
Let's first look at Jamal's solution:
Step 1: x^2 + 4x − 5 = 7
Step 2: x^2 + 4x − 5 − 7 = 7 − 7 (subtract 7 from both sides)
Step 2 (simplified): x^2 + 4x − 12 = 0
Now, let's check George's solution:
Step 1: x^2 + 4x − 5 = 7
Step 2: x^2 + 4x = 7 + 5 (add 5 to both sides)
Step 2 (simplified): x^2 + 4x = 12
Comparing the two solutions, we can see that George made a mistake in his step 2. He incorrectly added 5 to both sides of the equation, resulting in the incorrect equation x^2 + 4x = 12. This mistake could lead to incorrect roots or solutions for the quadratic equation.
On the other hand, Jamal's solution is accurate. He correctly subtracted 7 from both sides of the equation, resulting in the equivalent equation x^2 + 4x − 12 = 0. This equation is in the correct form for solving a quadratic equation using the Zero Product Property.
Therefore, we can conclude that Jamal's solution is accurate, and George's solution is inaccurate.
To determine which student's solution is accurate and which one is inaccurate, let's first review the Zero Product Property. The Zero Product Property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. In the context of solving a quadratic equation, this property tells us that if we have a quadratic expression equal to zero, we can set each factor of the expression equal to zero and solve for the variable.
Let's analyze Jamal's and George's work to see if their solutions align with the Zero Product Property.
Jamal's Solution:
1. x^2 + 4x - 5 = 7 (Given equation)
2. x^2 + 4x - 12 = 0 (Subtract 7 from both sides)
3. (x + 6)(x - 2) = 0 (Factoring the quadratic expression)
Since Jamal has correctly factored the quadratic expression into two binomial factors set equal to zero, he has followed the Zero Product Property correctly. Now, let's see if George's solution aligns with the property.
George's Solution:
1. x^2 + 4x - 5 = 7 (Given equation)
2. x^2 + 4x = 12 (Subtract 5 from both sides)
3. x(x + 4) = 12 (Factoring out the common factor from the quadratic expression)
At this point, George made an error. He only factored out one factor of 'x' from the quadratic expression, rather than fully factoring it. Therefore, his solution is not aligned with the Zero Product Property.
To summarize, the correct student's solution is accurate because they correctly factored the quadratic expression into two binomial factors set equal to zero. On the other hand, the incorrect student's solution is inaccurate because they failed to fully factor the quadratic expression, not satisfying the Zero Product Property.
To determine which student's solution is accurate, let's review their work and use the Zero Product Property to solve the quadratic equation x^2 + 4x - 5 = 7.
Jamal's Solution:
Step 1: x^2 + 4x - 5 = 7
Step 2: x^2 + 4x - 12 = 0
George's Solution:
Step 1: x^2 + 4x - 5 = 7
Step 2: x^2 + 4x = 12
To utilize the Zero Product Property, we want to find the values of x that make the equation equal to zero.
Jamal's Solution (x^2 + 4x - 12 = 0):
Now, we factor the quadratic equation:
(x - 2)(x + 6) = 0
The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero.
Therefore, x - 2 = 0 or x + 6 = 0. Solving for x, we get:
x = 2 or x = -6
So, Jamal's solution is accurate.
George's Solution (x^2 + 4x = 12):
Here, we need to subtract 12 from both sides of the equation:
x^2 + 4x - 12 = 0
Now we factor the quadratic equation:
(x - 2)(x + 6) = 0
Again, we use the Zero Product Property, which gives us:
x - 2 = 0 or x + 6 = 0
Solving for x, we get:
x = 2 or x = -6
Therefore, George's solution is also accurate.
Both Jamal and George's solutions are accurate since they both properly applied the Zero Product Property and obtained the same values for x.
In conclusion, both students' solutions are accurate, and there isn't an incorrect student's solution in this case.