Question
6x^2 - 7x +2=0
Kevin performed the first three seps but struggled in step 4 because he could not come up with a shared binomial in step 3. What change should he have made in step 3
6x^2 + (-3x) + (-4x) + 2=0
(6x^2+(-3x)) + (-4x+ 2) =0
3x(2x-1) + 2(-2x +1)=0
Use -2 as the GCF of the second grouping in step 3
Use -3x as the GCF of the first grouping in step 3
Change the grouping of pairs in step 2
Use different factors for -6x in step1
Kevin performed the first three seps but struggled in step 4 because he could not come up with a shared binomial in step 3. What change should he have made in step 3
6x^2 + (-3x) + (-4x) + 2=0
(6x^2+(-3x)) + (-4x+ 2) =0
3x(2x-1) + 2(-2x +1)=0
Use -2 as the GCF of the second grouping in step 3
Use -3x as the GCF of the first grouping in step 3
Change the grouping of pairs in step 2
Use different factors for -6x in step1
Answers
GPT-4o mini
Let's analyze the problem step by step to see where Kevin might have made a misstep and how he could improve his grouping.
We start with the equation:
\[ 6x^2 - 7x + 2 = 0 \]
### Step 1: Reorganizing
Kevin breaks down the middle term:
\[ 6x^2 + (-3x) + (-4x) + 2 = 0 \]
### Step 2: Grouping
He groups terms:
\[ (6x^2 + (-3x)) + (-4x + 2) = 0 \]
### Step 3: Factoring out GCF
When he reaches this point, he attempts to factor each group individually. The goal is to factor out a Greatest Common Factor (GCF) from each group.
- The first group, \( 6x^2 - 3x \), can factor out \( 3x \), so:
\[ 3x(2x - 1) \]
- The second group, \( -4x + 2 \), can be factored as:
\[ -2(2x - 1) \]
This gives:
\[ 3x(2x - 1) - 2(2x - 1) = 0 \]
### Step 4: Combining
Now, notice that both terms share a common binomial factor of \( (2x - 1) \):
\[ (3x - 2)(2x - 1) = 0 \]
So, if Kevin struggled in step 4 because he could not come up with a shared binomial, he should have factored the groups more effectively in step 3.
### Suggested Changes:
1. **Use GCF of the first grouping**: Change it to \( 3x(2x - 1) \).
2. **Use GCF of the second grouping**: Change it to \( -2(2x - 1) \).
3. **Combine the results**: Both groups can be combined as shown above to recognize the common factor.
### Answer Summary:
The change Kevin should have made in step 3 is to use \( 3x \) as the GCF of the first grouping and \(-2\) as the GCF of the second grouping, which would lead him to recognize the common factor of \( (2x - 1) \) for subsequent factoring.
This way, he will end up with \( (3x - 2)(2x - 1) = 0 \), making it easier to solve the equation.
We start with the equation:
\[ 6x^2 - 7x + 2 = 0 \]
### Step 1: Reorganizing
Kevin breaks down the middle term:
\[ 6x^2 + (-3x) + (-4x) + 2 = 0 \]
### Step 2: Grouping
He groups terms:
\[ (6x^2 + (-3x)) + (-4x + 2) = 0 \]
### Step 3: Factoring out GCF
When he reaches this point, he attempts to factor each group individually. The goal is to factor out a Greatest Common Factor (GCF) from each group.
- The first group, \( 6x^2 - 3x \), can factor out \( 3x \), so:
\[ 3x(2x - 1) \]
- The second group, \( -4x + 2 \), can be factored as:
\[ -2(2x - 1) \]
This gives:
\[ 3x(2x - 1) - 2(2x - 1) = 0 \]
### Step 4: Combining
Now, notice that both terms share a common binomial factor of \( (2x - 1) \):
\[ (3x - 2)(2x - 1) = 0 \]
So, if Kevin struggled in step 4 because he could not come up with a shared binomial, he should have factored the groups more effectively in step 3.
### Suggested Changes:
1. **Use GCF of the first grouping**: Change it to \( 3x(2x - 1) \).
2. **Use GCF of the second grouping**: Change it to \( -2(2x - 1) \).
3. **Combine the results**: Both groups can be combined as shown above to recognize the common factor.
### Answer Summary:
The change Kevin should have made in step 3 is to use \( 3x \) as the GCF of the first grouping and \(-2\) as the GCF of the second grouping, which would lead him to recognize the common factor of \( (2x - 1) \) for subsequent factoring.
This way, he will end up with \( (3x - 2)(2x - 1) = 0 \), making it easier to solve the equation.