Step 2: Write quadratic equation as the product of two factors: (x+6)(x−2)=0

Question

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. (jamal is correct)

Answer 1

To determine which student solved the quadratic equation correctly, let’s break down their work step by step and identify the mistakes.

1. Starting Equation: Both students started with the equation \(x^2 + 4x - 5 = 7\). This is correct.

2. Rearranging the Equation: To use the Zero Product Property, we need to set the equation to zero. So first, we should move 7 to the left side: \[ x^2 + 4x - 5 - 7 = 0 \] This simplifies to: \[ x^2 + 4x - 12 = 0 \]

3. Factoring the Quadratic:

   • Jamal's Work: He wrote the equation as \((x + 6)(x - 2) = 0\), which correctly factors to give: \[ x^2 + 4x - 12 \]
   • George's Work: He wrote the equation as \((x + 5)(x - 1) = 7\). This is incorrect since his factors do not match the correct equation.

4. Setting Factors to Zero:

   • Jamal: He set the correct factors equal to zero: \(x + 6 = 0\) and \(x - 2 = 0\).
   • George: He incorrectly wrote \(x + 5 = 0\) and \(x - 1 = 0\), which came from his incorrect factors.

5. Solving the Equations:

   • Jamal: Solving his equations gives \(x = -6\) and \(x = 2\).
   • George: Solving his equations gives \(x = -5\) and \(x = 1\).

Conclusion:

• Jamal's Solution is correct because he correctly factored the equation and used the Zero Product Property properly.
• George's Solution is inaccurate because his factors do not correctly represent the original equation after rearranging, and thus his final answers are incorrect.

By ensuring the original equation was set to zero and factored correctly, we can see why Jamal's solution is the accurate one.