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.

Answer 1

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

1. **Rearranging the Equation**:
Both students correctly start by rearranging the initial equation \( x^2 + 4x - 5 = 7 \) to set it equal to zero:
\[
x^2 + 4x - 5 - 7 = 0 \implies x^2 + 4x - 12 = 0 \quad \text{(correctly done by Jamal)}
\]

2. **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 \( x^2 + 4x - 12 \):
To factor \( x^2 + 4x - 12 \), 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 \( x^2 + 4x - 12 \):
George's factors, \( (x + 5)(x - 1) \), result in:
\[
x^2 - 1 + 5x - 5 = x^2 + 4x - 5
\]
That's incorrect for the equation at hand. Therefore, George's factorization is not valid.

3. **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 \( x^2 + 4x - 12 = 0 \) 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 \( x = 2 \).