Question
Jedida solved the quadratic equation x2−96=4x by factoring. Her work is shown below. At which step did Jedida first make a mistake, if at all?
x2−96=4x
Step 1: x2+4x−96=0 Add 4x to both sides.
Step 2: (x+12)(x−8)=0 Factor.
Step 3: x+12=0 or x−8=0 Use the Zero Product Property.
x=−12 or x=8
(1 point)
Responses
Step 3; she did not apply the Zero Product Property correctly.
Step 3; she did not apply the Zero Product Property correctly.
Step 1; she added 4x to both sides instead of subtracting.
Step 1; she added 4 x to both sides instead of subtracting.
Step 2; she did not factor the left side of the equation correctly.
Step 2; she did not factor the left side of the equation correctly.
She did not make any mistakes.
x2−96=4x
Step 1: x2+4x−96=0 Add 4x to both sides.
Step 2: (x+12)(x−8)=0 Factor.
Step 3: x+12=0 or x−8=0 Use the Zero Product Property.
x=−12 or x=8
(1 point)
Responses
Step 3; she did not apply the Zero Product Property correctly.
Step 3; she did not apply the Zero Product Property correctly.
Step 1; she added 4x to both sides instead of subtracting.
Step 1; she added 4 x to both sides instead of subtracting.
Step 2; she did not factor the left side of the equation correctly.
Step 2; she did not factor the left side of the equation correctly.
She did not make any mistakes.
Answers
GPT-4o mini
Jedida's work can be evaluated step by step:
1. **Step 1: \( x^2 - 96 = 4x \)**
She correctly moved \( 4x \) to the left side by adding \( 4x \) to both sides:
\[
x^2 - 4x - 96 = 0
\]
(Note: Here, it should have been \( x^2 - 4x - 96 = 0 \) instead of \( x^2 + 4x - 96 = 0 \).)
2. **Step 2:** She attempted to factor the equation:
The equation should have been:
\[
(x - 12)(x + 8) = 0
\]
But she factored it as:
\[
(x + 12)(x - 8) = 0
\]
This is incorrect as the factors don't match the quadratic.
3. **Step 3:** She applies the Zero Product Property:
Since the factors from Step 2 were incorrect, solving \( x + 12 = 0 \) and \( x - 8 = 0 \) would provide the wrong roots:
\[
x = -12 \quad \text{or} \quad x = 8
\]
This step is thus irrelevant due to the incorrect factorization.
Based on this analysis, the first mistake occurred in **Step 1**, where she incorrectly added \( 4x \), leading to the wrong quadratic to factor in Step 2.
So, the correct response is:
**Step 1; she added 4x to both sides instead of subtracting.**
1. **Step 1: \( x^2 - 96 = 4x \)**
She correctly moved \( 4x \) to the left side by adding \( 4x \) to both sides:
\[
x^2 - 4x - 96 = 0
\]
(Note: Here, it should have been \( x^2 - 4x - 96 = 0 \) instead of \( x^2 + 4x - 96 = 0 \).)
2. **Step 2:** She attempted to factor the equation:
The equation should have been:
\[
(x - 12)(x + 8) = 0
\]
But she factored it as:
\[
(x + 12)(x - 8) = 0
\]
This is incorrect as the factors don't match the quadratic.
3. **Step 3:** She applies the Zero Product Property:
Since the factors from Step 2 were incorrect, solving \( x + 12 = 0 \) and \( x - 8 = 0 \) would provide the wrong roots:
\[
x = -12 \quad \text{or} \quad x = 8
\]
This step is thus irrelevant due to the incorrect factorization.
Based on this analysis, the first mistake occurred in **Step 1**, where she incorrectly added \( 4x \), leading to the wrong quadratic to factor in Step 2.
So, the correct response is:
**Step 1; she added 4x to both sides instead of subtracting.**