Use the tables to answer the question.

Question

Use the tables to answer the 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.

GPT-4o mini · Answer

To determine which student is incorrect, let's carefully analyze both their approaches to solving the quadratic equation $x^{2} + 4 x - 5 = 7$ .

Rearranging the Equation: Both students correctly start by rearranging the initial equation $x^{2} + 4 x - 5 = 7$ to set it equal to zero: $x^{2} + 4 x - 5 - 7 = 0 ⟹ x^{2} + 4 x - 12 = 0 (correctly done by Jamal)$
Factoring:
- Jamal's factors: He factored the equation as $(x + 6) (x - 2) = 0$ .
- George's factors: He incorrectly wrote the factors as $(x + 5) (x - 1) = 0$ .

Let's verify the correct factors of $x^{2} + 4 x - 12$ : To factor $x^{2} + 4 x - 12$ , we look for two numbers that multiply to $- 12$ (the constant term) and add up to $4$ (the coefficient of $x$ ). The numbers $6$ and $- 2$ satisfy this requirement: $(x + 6) (x - 2) = 0$ So, Jamal's factoring is correct, and he finds the solutions $x = - 6$ and $x = 2$ .

Now, we can also verify the factors that George provided for the original equation $x^{2} + 4 x - 12$ : George's factors, $(x + 5) (x - 1)$ , result in: $x^{2} - 1 + 5 x - 5 = x^{2} + 4 x - 5$ That's incorrect for the equation at hand. Therefore, George's factorization is not valid.

Solutions:
- Jamal finds the solutions as $x = - 6$ and $x = 2$ .
- George finds the incorrect solutions as $x = - 5$ and $x = 1$ .

Conclusion: George is the incorrect student. His initial factorization of the quadratic $x^{2} + 4 x - 12 = 0$ was erroneous, leading him to the wrong solutions. The correct roots are obtained from Jamal’s accurate factorization $(x + 6) (x - 2) = 0$ , leading to $x = - 6$ and $x = 2$ .