Question

Kevin is solving 6x2−7x+2=0
by grouping. He performed the first three steps as shown below. However, he struggles to complete Step 4 because he could not come up with a shared binomial in Step 3. What change should Kevin make to arrive at a shared binomial in Step 3?

Step 1: 6x2+(−3x)+(−4x)+2=0


Step 2: (6x2+(−3x))+(−4x+2)=0


Step 3: 3x(2x−1)+2(−2x+1)=0


(1 point)
Responses

Change the grouping of pairs in Step 2.
Change the grouping of pairs in Step 2.

Use −3x
as the GCF of the first grouping in Step 3.
Use negative 3 x as the GCF of the first grouping in Step 3.

Use −2
as the GCF of the second grouping in Step 3.
Use negative 2 as the GCF of the second grouping in Step 3.

Use different factors for −7x
in Step 1.

Answer 1

To help Kevin arrive at a shared binomial in Step 3, he needs to adjust the way he's factoring out the greatest common factors (GCF) in his groupings.

In the current Step 3, he factored \(3x(2x - 1)\) from the first term, which is correct, but he factored \(2(-2x + 1)\) from the second term which should instead yield a different binomial.

To achieve a common binomial factor, Kevin should consider changing the GCF of the second grouping in Step 3.

Using 2 as the GCF of the second grouping, we recognize:
- From \(-4x + 2\), he can factor as:
\[ -2(2x - 1)\]

Thus, both groups will lead to a common binomial of \(2x - 1\).

Therefore, the best response for Kevin to create a shared binomial in Step 3 is:

**Use −2 as the GCF of the second grouping in Step 3.**