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 simple and intended for 9th grade.

Answer 1

To solve the quadratic equation \(x^2 + 4x - 5 = 7\), we first need to move all terms to one side of the equation to set it equal to 0. This gives us:

1. **Correct Step:** Rewrite the equation:
\[x^2 + 4x - 5 - 7 = 0\]
Which simplifies to:
\[x^2 + 4x - 12 = 0\]

Now, let’s compare the solutions from George and Jamal.

### George's Solution:
- He started with the equation **x² + 4x - 5 = 7** but did not set it equal to zero correctly. He wrongly wrote it as:
\[(x+5)(x-1)=7\]
This is not correct because he didn’t actually simplify the equation properly; it should be set to zero first.

### Jamal's Solution:
- He correctly started with:
\[x^2 + 4x - 12 = 0\]
- Then, he factored it correctly as:
\[(x + 6)(x - 2) = 0\]
- He then correctly used the Zero Product Property, setting each factor to zero:
\[x + 6 = 0\] and \[x - 2 = 0\]
- Solving these gives:
\[x = -6\] and \[x = 2\]

### Conclusion:
- **Jamal** is correct because he followed the right steps of solving a quadratic equation, starting with setting it equal to 0 and factoring correctly.
- **George** is incorrect because he did not rewrite the equation correctly to set it to zero, and thus his factors and solutions are wrong.

In summary, the key difference is that Jamal set the equation to zero first and factored it correctly, while George did not.