Hyung-eun is solving the quadratic equation x^2−4x−12=0

Question

by grouping. Her work is shown below. At which step does Hyung-eun first make an error?

x^2−4x−12=0


Step1: x^2+(−6x)+2x−12=0


Step 2: (x^2+(−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 4
Step 4

Step 3
Step 3

Step 2
Step 2

Step 1
Step 1

Answer 1

To find where Hyung-eun makes an error, let's analyze each step of her work.

The original equation is: \[ x^2 - 4x - 12 = 0 \]

Step 1: \( x^2 + (-6x) + 2x - 12 = 0 \)

• At this step, Hyung-eun tries to split \(-4x\) into \((-6x) + (2x)\). This splitting is incorrect because \(-6 + 2 = -4\) does not help in solving the equation and is an unusual way to factor the quadratic. This technically could be the first step where she introduces a possible error in grouping.

Step 2: \( (x^2 + (-6x)) + (2x - 12) = 0 \)

• The grouping is consistent with her previous step; however, the way she split the middle term makes factoring incorrectly lead to complications.

Step 3: \( x(x + (-6)) + 2(x - 6) = 0 \)

• This grouping is incorrect. The factorization of \(x^2 + (-6x)\) is faulty. The component \((2x - 12)\) does not correctly factor into \(2(x - 6)\) since the extracted term should have been \(2(x + 6)\) leading to an already incorrect resolution of factors.

Step 4: \( (x + 6)(x - 2) = 0 \)

• At this point, she incorrectly factors \(x(x - 6) + 2(x - 6)\) into \((x + 6)(x - 2)\). The factorization does not match any known valid factorizations corresponding to the original equation.

The first significant mistake occurs in Step 1, during the incorrect splitting of the middle term. Therefore, the answer is:

Step 1