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.
, 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.
Answers
GPT-4o mini
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.
### 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.