Kevin's steps for solving the equation \(6x^2−7x+2=0\) show that he is trying to factor by grouping, but he needs to ensure that the terms are arranged in a way that allows for a shared binomial factor.
In Kevin's Step 1, he rewrote \(-7x\) as \(-3x - 4x\). However, he can make a more effective choice in splitting the middle term, which will help him achieve the shared binomial.
To arrive at a shared binomial in Step 3, Kevin should split \(-7x\) in a way that allows for matching coefficients. Specifically, if we factor the quadratic directly:
- Find two numbers that multiply to the product of the coefficient of \(x^2\) (which is 6) and the constant term (which is 2):
- The product is \(6 \cdot 2 = 12\).
- Find two numbers that sum to \(-7\) (the coefficient of \(x\)) and multiply to \(12\):
- These numbers are \(-3\) and \(-4\), which Kevin correctly used, but let's consider how we might group them more effectively.
Instead of \(-3x\) and \(-4x\):
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Rearrange \(6x^2 - 4x - 3x + 2 = 0\) to help see if there's a common factor:
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Write the equation as: \[ 6x^2 - 4x - 3x + 2 = 0 \]
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Then group them as: \[ (6x^2 - 4x) + (-3x + 2) = 0 \]
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Now factor each part: \[ 2x(3x - 2) - 1(3x - 2) = 0 \]
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Now, in Step 3, you will see the shared binomial:
- \( (2x - 1)(3x - 2) = 0 \)
Thus, to arrive at a shared binomial in Step 3, Kevin could revise his grouping or arrangement of terms in Step 1 to \(6x^2 - 4x - 3x + 2 = 0\), which would lead to the correct shared binomial during the factoring process.
So, the change Kevin should make is to split \(-7x\) into \(-4x\) and \(-3x\) instead. This will allow him to create the shared binomial in Step 3 effectively.
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