Kevin is solving 6x^2−7x+2=0

Question

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: 6x^2+(−3x)+(−4x)+2=0


Step 2: (6x^2+(−3x))+(−4x+2)=0


Step 3: 3x(2x−1)+2(−2x+1)=0

(1 point)
Responses

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.
Use different factors for negative 7 x in Step 1.

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.

Change the grouping of pairs in Step 2.

Answer 1

To arrive at a shared binomial in Step 3, Kevin should consider using different factors for \(-7x\) in Step 1.

In Step 1, he wrote \(6x^2 + (-3x) + (-4x) + 2 = 0\). The correct way to factor \(-7x\) into two numbers that multiply to \(6 \times 2 = 12\) and add up to \(-7\) is to use \(-3\) and \(-4\), which he did. However, these particular choices didn't lead to a shared binomial in Step 3.

Instead, Kevin could use the pair \(-6x\) and \(-1x\) instead of \(-3x\) and \(-4x\). This would change Step 1 to:

\[6x^2 + (-6x) + (-1x) + 2 = 0\]

That way, when performing grouping in Step 2 and factoring in Step 3, he could potentially get shared binomials.

So, the best option from the choices would be:

Use different factors for \(-7x\) in Step 1.