Use the tables to answer the question.

Question

When trying to solve the quadratic equation x2+4x−5=7
, Jamal and George each provided their work in the tables.

Jamal

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: Set equation = 0 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

Which student is incorrect? Explain how you know that his solution is inaccurate.

(2 points)

Answer 1

To determine which student is incorrect, we must carefully analyze each student's work to identify any errors in their calculations or factoring.

Jamal's Work:

1. Step 1: Correctly set the equation to 0: \[ x^2 + 4x - 5 - 7 = 0 \Longrightarrow x^2 + 4x - 12 = 0 \] (This is correct).

2. Step 2: Factoring \(x^2 + 4x - 12 = 0\): \[ (x + 6)(x - 2) = 0 \] (This is also correct since \(6 \times (-2) = -12\) and \(6 + (-2) = 4\)).

3. Step 3: Setting each factor equal to 0: \[ x + 6 = 0 \quad \text{and} \quad x - 2 = 0 \]

4. Step 4: Solving for \(x\): \[ x = -6 \quad \text{and} \quad x = 2 \]

George's Work:

1. Step 1: Did not correctly set the equation to 0: \[ x^2 + 4x - 5 = 7 \Longrightarrow x^2 + 4x - 12 = 0 \]

   • This step should match Jamal’s.

2. Step 2: Incorrectly factored the equation: \[ (x + 5)(x - 1) = 7 \]

   • This factored form is incorrect. The proper factors should result from the quadratic \(x^2 + 4x - 12\), which Jamal found correctly.

3. Step 3: Setting factors equal to 0 doesn’t apply since the factors are wrong.

4. Step 4: The solutions \(x = -5\) and \(x = 1\) derived from George’s incorrect factorization are also invalid.

Conclusion:

George is incorrect. His initial factoring of the quadratic equation is flawed, as he did not provide the correct factors for \(x^2 + 4x - 12\). Instead of finding the accurate factors and roots, he has derived incorrect answers from incorrect premises. Therefore, the solutions he provided are not valid.