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Kevin is solving 6x^2−7x+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: 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

A) Use different factors for −7x in Step 1.

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

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

D) Change the grouping of pairs in Step 2.

7 answers

To arrive at a shared binomial in Step 3, Kevin should focus on grouping the terms in a way that common factors can be identified effectively. Let's analyze the options given:

  1. A) Use different factors for −7x in Step 1.
    This option suggests changing the way the middle term (−7x) is split, which could potentially lead to a shared binomial in Step 3.

  2. B) Use −2 as the GCF of the second grouping in Step 3.
    This would not necessarily help since the grouping itself is already leading to a different structure (−2 would be factoring out from a group that is currently not leading to shared binomials).

  3. C) Use −3x as the GCF of the first grouping in Step 3.
    This option wouldn't directly help in achieving a shared binomial but would complicate the existing grouping.

  4. D) Change the grouping of pairs in Step 2.
    Changing the grouping might allow different terms to be combined, potentially creating a scenario for a shared binomial.

Given the analysis above, the best option that would help Kevin arrive at a shared binomial in Step 3 is:

A) Use different factors for −7x in Step 1.

This allows for potential combinations that can lead to a common factor in subsequent steps. Therefore, Kevin should try splitting −7x into different factors instead of −3x and −4x.

That's incorrect

I apologize for the confusion. Let's re-evaluate the steps and the options given in detail.

Kevin's current sequence is:

Step 1: \(6x^2 + (-3x) + (-4x) + 2 = 0\)
Step 2: \((6x^2 + (-3x)) + (-4x + 2) = 0\)
Step 3: \(3x(2x - 1) + 2(-2x + 1) = 0\)

In Step 3, Kevin has the expression \(3x(2x - 1) + 2(-2x + 1) = 0\), but the second binomial, \(-2x + 1\), does not help in arriving at a shared binomial.

To achieve a shared binomial, Kevin needs to properly group and factor.

The options provided are:

A) Use different factors for −7x in Step 1.
B) Use −2 as the GCF of the second grouping in Step 3.
C) Use −3x as the GCF of the first grouping in Step 3.
D) Change the grouping of pairs in Step 2.

Let's reconsider the options:

  • A) Use different factors for −7x in Step 1.
    This option suggests a different factorization for \(-7x\). However, it would still lead back to a similar situation in factoring.

  • B) Use −2 as the GCF of the second grouping in Step 3.
    This could lead to better simplification from the current factor \(2(-2x + 1)\), but it does not directly result in a shared binomial.

  • C) Use −3x as the GCF of the first grouping in Step 3.
    This yet again complicates the overall grouping rather than simplifying it to a shared binomial form.

  • D) Change the grouping of pairs in Step 2.
    This option allows for a different approach. If Kevin pairs the terms differently, he could create more favorable combinations that would lead to a common binomial.

After evaluating the options again, the most appropriate choice for Kevin to achieve a shared binomial in Step 3 would be:

D) Change the grouping of pairs in Step 2.

This allows Kevin to reformulate the terms in a manner conducive to finding a shared binomial in later steps. Thank you for your patience, and I hope this clarification helps!

That's still incorrect

I apologize for the oversight. Let’s carefully analyze Kevin’s steps and the options once more.

Kevin's steps are as follows:

Step 1: \(6x^2 + (-3x) + (-4x) + 2 = 0\)
Step 2: \((6x^2 + (-3x)) + (-4x + 2) = 0\)
Step 3: \(3x(2x - 1) + 2(-2x + 1) = 0\)

As you pointed out, the expression in Step 3, \(3x(2x - 1) + 2(-2x + 1) = 0\), does not yield a shared binomial factor.

To achieve a shared binomial factor, we need to revisit how Kevin split the middle term and what he factored out in his additional steps.

Let's analyze the options again:

  1. A) Use different factors for −7x in Step 1.
    By using different factors, which should be \(-3x\) and \(-4x\) to split \(-7x\), we might not get better results and could lead to more complications instead of clarifying how to reach a shared binomial.

  2. B) Use −2 as the GCF of the second grouping in Step 3.
    If we try to factor out \(-2\) from \(2(-2x + 1)\), this doesn’t lead us toward a shared binomial, and may complicate the process.

  3. C) Use −3x as the GCF of the first grouping in Step 3.
    This does not produce a useful shared factor, as it would complicate the equations further.

  4. D) Change the grouping of pairs in Step 2.
    Changing the grouping could indeed yield a better outcome, but if we analyze the factual breakdown of the entire polynomial, we need to see if differing configurations or splits will yield results in later steps.

Reverting to the Expression

Now, let’s step back and focus on the completed factorization. The goal is to rewrite or regroup in a way that keeps the expression neat.

In fact, if Kevin correctly grouped the original equation:

For \(6x^2 - 7x + 2\):

He can factor it directly by searching for pairs that yield a similar coefficient.

A feasible rearrangement is:

Step 1: \(6x^2 - 3x - 4x + 2 = 0\) (current method).
But if we reconsider the binomial split that correctly adds and multiplies correctness with simplicity, the key pairs for Step 1 should be revised.

Thus correctly pairing yields values that reveal commonality.

Correct Answer

The best match from analysis reveals that:

A is the most valid choice given what needs to change for Kevin to factor based on constructing relevant factors to match coefficients.

So explicitly, using different factors for \(-7x\) will clarify how to yield a shared binomial in future steps---a key step of clarification and factor match.

Thank you and my sincere apologies for the confusion in previous details!

Try again.

I apologize for the confusion in the previous responses. Let’s analyze Kevin’s steps and the available options more clearly to find the correct answer.

Kevin's Steps:

  1. Step 1: \(6x^2 + (-3x) + (-4x) + 2 = 0\)
  2. Step 2: \((6x^2 + (-3x)) + (-4x + 2) = 0\)
  3. Step 3: \(3x(2x - 1) + 2(-2x + 1) = 0\)

Analyzing Step 3

In Step 3, Kevin factored:

  • From \(6x^2 - 3x\), he correctly factored out \(3x\) to get \(3x(2x - 1)\).
  • However, from the expression \(-4x + 2\), he factored out \(2\) and incorrectly wrote it as \(2(-2x + 1)\) instead of \(2(2 - 2x)\).

Desired Scenario

Kevin needs a shared binomial factor when he regroup the expression. Therefore, the current factoring does not yield the necessary shared term which would be helpful in the solution.

Options:

A) Use different factors for −7x in Step 1.
B) Use −2 as the GCF of the second grouping in Step 3.
C) Use −3x as the GCF of the first grouping in Step 3.
D) Change the grouping of pairs in Step 2.

Evaluation of Options

  1. A) Use different factors for −7x in Step 1.
    While splitting \(-7x\) into different pairs could be helpful, it may not necessarily lead to a shared binomial.

  2. B) Use −2 as the GCF of the second grouping in Step 3.
    If Kevin used \(-2\) as the GCF of the second grouping, he could factor \(2\) out to yield a clearer shared binomial.

  3. C) Use −3x as the GCF of the first grouping in Step 3.
    This doesn’t improve the chances of revealing a shared binomial since it alters the original expression's structure.

  4. D) Change the grouping of pairs in Step 2.
    Changing the grouping might help but won't directly influence the simplification achieved in Step 3.

Correct Answer

The best choice for Kevin to arrive at a shared binomial in Step 3 is:

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

This would lead to a proper simplification, revealing the shared binomial that he is looking for in order to move forward with the problem.

Thank you for your patience, and I hope this answer correctly addresses the question!