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.

Answer 1

To determine which student's solution is accurate, we need to analyze their work step-by-step.

Jamal's Solution:

1. Equation Set to 0:

   • Jamal correctly starts with the equation \(x^2 + 4x - 12 = 0\), but this appears to be a mistake. It should be \(x^2 + 4x - 5 - 7 = 0\), which simplifies to \(x^2 + 4x - 12 = 0\).

2. Factoring:

   • Jamal factors \(x^2 + 4x - 12\) as \((x + 6)(x - 2) = 0\). This is a correct factorization since expanding \((x + 6)(x - 2)\) gives \(x^2 + 4x - 12\).

3. Setting Factors to 0:

   • He sets each factor to zero: \(x + 6 = 0\) and \(x - 2 = 0\).

4. Solving Each Equation:

   • He correctly finds the solutions \(x = -6\) and \(x = 2\), which are valid solutions for his derived equation.

George's Solution:

1. Equation Set to 0:

   • George starts with the equation \(x^2 + 4x - 5 = 7\), which he then incorrectly rearranges to set it to zero: \(x^2 + 4x - 5 - 7 = 0\) simplifies to \(x^2 + 4x - 12 = 0\).

2. Factoring:

   • George claims to factor \(x^2 + 4x - 5\) incorrectly as \((x + 5)(x - 1) = 7\), which is incorrect in two ways. First, he should be factoring the set-to-zero version of the equation \(x^2 + 4x - 12 = 0\) and not the original equation \(x^2 + 4x - 5\). Secondly, \((x + 5)(x - 1)\) expands to \(x^2 + 4x - 5\) and does not account for the 7 added to the left side.

3. Setting Factors to 0:

   • George sets \(x + 5 = 0\) and \(x - 1 = 0\), which is based on his incorrect factorization.

4. Solving Each Equation:

   • He correctly solves these equations to get \(x = -5\) and \(x = 1\), but these solutions are based on the wrong factorization and therefore do not correspond to the original equation.

Conclusion:

• Correct Solution: Jamal's final answers are \(x = -6\) and \(x = 2\) based on a correct derivation of the transformed equation. Hence, his solution is accurate for the quadratic equation \(x^2 + 4x - 12 = 0\).
• Incorrect Solution: George's factorization is incorrect for the equation because he mistakenly factors the wrong quadratic. His solutions do not correspond to the original problem and are therefore inaccurate.

Thus, Jamal's approach adheres properly to the steps required to solve the given quadratic equation, while George's incorrect transformation and factorization lead to invalid answers.