Let's review each step to see where the first error occurs:
Step 1: The equation \( x^2 - 4x - 12 = 0 \) is split into \( x^2 - 6x + 2x - 12 = 0 \).
This step is valid. The choice of -6 and +2 as coefficients for x gives us the original -4x when combined.
Step 2: Grouping the terms as \( (x^2 - 6x) + (2x - 12) = 0 \).
This step is also valid. It's simply regrouping the terms to prepare for factoring by grouping.
Step 3: Factoring out the common terms gives us \( x(x - 6) + 2(x - 6) = 0 \).
This step seems valid as well. The factoring by grouping is done correctly, and we can see the common factor \( x - 6 \).
Step 4: Factoring out the common binomial \( (x - 6) \), the equation becomes \( (x + 2)(x - 6) = 0 \).
Here we find the error. Hyung-eun incorrectly factored the equation as \( (x + 6)(x - 2) = 0 \), which is not correct. It should be \( (x + 2)(x - 6) = 0 \).
Step 5: Solving for x, she gets \( x = -6 \) or \( x = 2 \).
The solutions provided here don't match the factors \( (x + 2)(x - 6) \). The correct solutions should be \( x = -2 \) or \( x = 6 \).
The correct response is:
Step 4
Hyung-eun is solving the quadratic equation x2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?
x2−4x−12=0
Step1: x2+(−6x)+2x−12=0
Step 2: (x2+(−6x))+(2x−12)=0
Step 3: x(x+(−6))+2(x−6)=0
x(x−6)+2(x−6)=0
Step 4: (x+6)(x−2)=0
Step 5: x+6=0 or x−2=0
x=−6 or x=2
(1 point)
Responses
Step 2
Step 2
Step 3
Step 3
Step 4
Step 4
Step 1
Step 1
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