When trying to solve the quadratic equation x2+4x−5=7

Question

, 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.

Answer 1

To determine which student's solution is incorrect, let's examine both Jamal's and George's work step by step.

Original Equation:

The original equation given is: \[ x^2 + 4x - 5 = 7 \]

Step 1 for Both: Set Equation to 0

Both students need to rearrange the equation to have it equal to zero. This means moving 7 to the left side: \[ x^2 + 4x - 5 - 7 = 0 \] This simplifies to: \[ x^2 + 4x - 12 = 0 \]

Jamal's Step 1 is correct.

Now for George's Step 1, he incorrectly stated: \[ x^2 + 4x - 5 = 7 \] and did not set it to 0 correctly.

Analysis of Steps:

• Jamal correctly wrote the equation as: \[ x^2 + 4x - 12 = 0 \]

• George did not set the equation to zero correctly and continued from his incorrect equation.

Step 2: Factoring

Now let's see their factorization.

1. Jamal's Factorization: He factored \( x^2 + 4x - 12 \): \[ (x + 6)(x - 2) = 0 \]

   To confirm:

   • The factors of -12 that add up to 4 are indeed 6 and -2.

2. George's Factorization: He stated: \[ (x + 5)(x - 1) = 7 \] This is inherently incorrect since it does not represent a correct factorization of the original equation. The left-hand side was supposed to be 0.

Step 3: Setting Factors to Zero

• Jamal set his factors correctly as: \[ x + 6 = 0 \quad \text{and} \quad x - 2 = 0 \]

  From these, he correctly found the solutions: \[ x = -6 \quad \text{and} \quad x = 2 \]

• George's step does not apply since his factors (incorrectly set to equal 7) were based on the wrong equation.

Conclusion:

Jamal is correct in his solving of the quadratic equation. He ended up with correct solutions of \( x = -6 \) and \( x = 2 \).

George is incorrect; he did not rearrange the original equation properly in Step 1, leading to incorrect factorization and incorrect solutions. Therefore, his method is inaccurate from the very beginning.