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?

Question

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 1
Step 1

Step 4
Step 4

Step 2
Step 2

Step 3

Answer 1

Let's analyze the steps provided by Hyung-eun to identify where the error occurred in her work while solving the quadratic equation \( x^2 - 4x - 12 = 0 \) by grouping.

1. Original equation: \[ x^2 - 4x - 12 = 0 \]

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

   • At this step, Hyung-eun incorrectly decomposed \(-4x\) into \(-6x\) and \(2x\). This is the first error. The correct decomposition of \(-4x\) should have been \(-6x + 2x\) because that should sum to \(-4x\).

3. Step 2: \[ (x^2 + (-6x)) + (2x - 12) = 0 \]

   • While this grouping is consistent with her prior step, the earlier mistake in Step 1 carries through.

4. Step 3: \[ x(x + (-6)) + 2(x - 6) = 0 \]

   • This step attempts to factor; however, already the factors are incorrect because of the earlier mistake.

5. Step 4: \[ (x + 6)(x - 2) = 0 \]

   • The factors presented here are incorrect due to the errors in prior steps.

6. Step 5:

   • This step yields the incorrect roots, where she claims \( x = -6 \) or \( x = 2 \).

Since the first mistake occurred in Step 1, where \( -4x \) was incorrectly broken into \( -6x + 2x \), the answer is:

Step 1