Jamal:

Question

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. Please make your answer simply written and intended for 9th grade.

Answer 1

To determine whose solution is accurate, let's take a closer look at their work step by step.

Jamal's Work:

1. He correctly started with the equation \(x^2 + 4x - 12 = 0\).
2. He factored it as \((x + 6)(x - 2) = 0\). This means he's stated that the equation can be broken down into these two factors, which is correct.
3. He set each factor equal to 0: \(x + 6 = 0\) and \(x - 2 = 0\).
4. Solving these gives \(x = -6\) and \(x = 2\).

This process is correct, and Jamal properly applied the Zero Product Property by setting each factor equal to zero.

George's Work:

1. George started the equation: \(x^2 + 4x - 5 = 7\).
2. He incorrectly wrote \((x + 5)(x - 1) = 7\) as the factored form. However, this is not correct because we need to set the equation to 0 first. The correct step should be:
   • Move 7 to the left side to get \(x^2 + 4x - 5 - 7 = 0\) or \(x^2 + 4x - 12 = 0\).
   • Once he has \(x^2 + 4x - 12 = 0\), he would then factor it correctly as \((x + 6)(x - 2) = 0\) just like Jamal did.
3. Because his factored form is incorrect, the equations he set equal to 0, \(x + 5 = 0\) and \(x - 1 = 0\), are also not correct.
4. Consequently, when he solves these and finds \(x = -5\) and \(x = 1\), these answers are not valid solutions to the original equation.

In conclusion, Jamal's solution is accurate because he correctly factored the equation after setting it to zero and applied the Zero Product Property correctly. George's solution is inaccurate because he did not start with the equation in the correct form and made a mistake in the factoring process.