Step 1: Set equation = 0: x2+4x−12=0

Question

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 which student's solution is accurate, we need to look closely at their work for both George and Jamal.

Jamal's steps:

1. He starts with the equation: \( x^2 + 4x - 5 = 7 \).
2. He doesn't first move everything to one side to set the equation to 0. Instead, he jumps to writing it as \( (x + 6)(x - 2) = 0 \), which is not derived correctly from the original equation.
3. This factorization is incorrect, that means his foundation to use the Zero Product Property is flawed.

George's steps:

1. He starts with the equation: \( x^2 + 4x - 5 = 7 \).
2. Correctly, he should first rearrange the equation to set it equal to 0: \[ x^2 + 4x - 5 - 7 = 0 , \Rightarrow , x^2 + 4x - 12 = 0. \]
3. After moving terms around correctly, George writes \( (x + 6)(x - 2) = 0 \), which is accurate because the original equation was adjusted correctly.

Comparison:

• Jamal's mistake: He didn't set the equation equal to 0 before factoring, which is crucial for using the Zero Product Property correctly. His factors \( (x + 6)(x - 2) \) are not correct based on what the equation should equal.

• George's correctness: He correctly set the equation to 0 and came up with valid factors. Thus, when he uses the Zero Product Property, his solutions of \( x = -6 \) and \( x = 2 \) are based on a valid equation.

In conclusion, George's solution is accurate because he correctly set the equation to 0 and factored it properly, while Jamal's attempt is inaccurate because he skipped the critical step of setting the equation to 0 before factoring.