Question
Jason solved the quadratic equation x2−4x−12=0
by factoring. His work is shown below.
Step 1: x2−4x−12=0
Step 2: (x-6)(x+2)=0
Step 3: x - 6 = 0, x + 2 = 0
Step 4: x = 6 or x = 2
Step 5: {6,2}
Jason plugged his solutions into the original equation, and found that they did not work. In which step did he first make a mistake?
by factoring. His work is shown below.
Step 1: x2−4x−12=0
Step 2: (x-6)(x+2)=0
Step 3: x - 6 = 0, x + 2 = 0
Step 4: x = 6 or x = 2
Step 5: {6,2}
Jason plugged his solutions into the original equation, and found that they did not work. In which step did he first make a mistake?
Answers
GPT-4o mini
Let's analyze Jason's work step-by-step:
1. **Step 1**: The original equation \( x^2 - 4x - 12 = 0 \) is correct.
2. **Step 2**: He factored the equation as \( (x - 6)(x + 2) = 0 \). To verify this, we can expand \( (x - 6)(x + 2) \):
\[
(x - 6)(x + 2) = x^2 + 2x - 6x - 12 = x^2 - 4x - 12
\]
This step is actually correct.
3. **Step 3**: From \( (x - 6)(x + 2) = 0 \), he correctly sets each factor to zero:
- \( x - 6 = 0 \) gives \( x = 6 \)
- \( x + 2 = 0 \) gives \( x = -2 \)
Here, he made a mistake in writing down the second solution. It should be \( x = -2 \), not \( x = 2 \).
4. **Step 4**: He states \( x = 6 \) or \( x = 2 \). This is incorrect because one of the solutions is wrong.
5. **Step 5**: Jason concludes that the solutions are \( \{6, 2\} \), which is incorrect.
Therefore, Jason's first mistake occurred in **Step 3**, where he incorrectly stated the second solution as \( x = 2 \) instead of the correct solution \( x = -2 \).
1. **Step 1**: The original equation \( x^2 - 4x - 12 = 0 \) is correct.
2. **Step 2**: He factored the equation as \( (x - 6)(x + 2) = 0 \). To verify this, we can expand \( (x - 6)(x + 2) \):
\[
(x - 6)(x + 2) = x^2 + 2x - 6x - 12 = x^2 - 4x - 12
\]
This step is actually correct.
3. **Step 3**: From \( (x - 6)(x + 2) = 0 \), he correctly sets each factor to zero:
- \( x - 6 = 0 \) gives \( x = 6 \)
- \( x + 2 = 0 \) gives \( x = -2 \)
Here, he made a mistake in writing down the second solution. It should be \( x = -2 \), not \( x = 2 \).
4. **Step 4**: He states \( x = 6 \) or \( x = 2 \). This is incorrect because one of the solutions is wrong.
5. **Step 5**: Jason concludes that the solutions are \( \{6, 2\} \), which is incorrect.
Therefore, Jason's first mistake occurred in **Step 3**, where he incorrectly stated the second solution as \( x = 2 \) instead of the correct solution \( x = -2 \).