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Which of the answer choices gives a correct grouping of the following quadratic?

[tex]\[6x^2 + 7x - 5\][/tex]

Select the correct answer below:

A. [tex]\(6x^2 - 4x + 11x - 5\)[/tex]

B. [tex]\(6x^2 - 2x + 9x - 5\)[/tex]

C. [tex]\(6x^2 - 6x + 9x - 9\)[/tex]

D. [tex]\(6x^2 - 3x + 10x - 5\)[/tex]

Sagot :

GinnyAnswer
To find the correct grouping for the quadratic expression [tex]\(6x^2 + 7x - 5\)[/tex], we need to break down the middle term ([tex]\(7x\)[/tex]) into two terms such that the product of the coefficients of the two new terms is equal to the product of the coefficient of [tex]\(x^2\)[/tex] (which is 6) and the constant term (which is -5). In other words, we need to find two numbers [tex]\((a\)[/tex] and [tex]\(b)\)[/tex] such that:

1. [tex]\(a \cdot b = 6 \cdot (-5) = -30\)[/tex]
2. [tex]\(a + b = 7\)[/tex]

Let's go through the choices to identify the correct one:

1. [tex]\(6x^2 - 4x + 11x - 5\)[/tex]

- Group terms: [tex]\(6x^2 - 4x + 11x - 5\)[/tex]
- Combine like terms to check: [tex]\((6x^2 - 4x) + (11x - 5)\)[/tex]
- This simplifies to: [tex]\(6x^2 + 7x - 5\)[/tex]
- Conclusion: Correct grouping

2. [tex]\(6x^2 - 2x + 9x - 5\)[/tex]

- Group terms: [tex]\(6x^2 - 2x + 9x - 5\)[/tex]
- Combine like terms to check: [tex]\((6x^2 - 2x) + (9x - 5)\)[/tex]
- This simplifies to: [tex]\(6x^2 + 7x - 5\)[/tex]
- Conclusion: Correct grouping

3. [tex]\(6x^2 - 6x + 9x - 9\)[/tex]

- Group terms: [tex]\(6x^2 - 6x + 9x - 9\)[/tex]
- Combine like terms to check: [tex]\((6x^2 - 6x) + (9x - 9)\)[/tex]
- This simplifies to: [tex]\(6x^2 + 3x - 9\)[/tex]
- Conclusion: Incorrect grouping

4. [tex]\(6x^2 - 3x + 10x - 5\)[/tex]

- Group terms: [tex]\(6x^2 - 3x + 10x - 5\)[/tex]
- Combine like terms to check: [tex]\((6x^2 - 3x) + (10x - 5)\)[/tex]
- This simplifies to: [tex]\(6x^2 + 7x - 5\)[/tex]
- Conclusion: Correct grouping

Thus, the correct answers are:

- [tex]\(6x^2 - 4x + 11x - 5\)[/tex]
- [tex]\(6x^2 - 2x + 9x - 5\)[/tex]
- [tex]\(6x^2 - 3x + 10x - 5\)[/tex]

So, the valid correct answer choices that gives a correct grouping of the quadratic [tex]\(6x^2 + 7x - 5\)[/tex] are:

- [tex]\(6x^2 - 4x + 11x - 5\)[/tex]
- [tex]\(6x^2 - 2x + 9x - 5\)[/tex]
- [tex]\(6x^2 - 3x + 10x - 5\)[/tex]

However, based on the examination, all the given answers except [tex]\(6x^2 - 6x + 9x - 9\)[/tex] are feasible, so we can conclude:

The correct answers are:
1. [tex]\(6x^2 - 4x + 11x - 5\)[/tex]
2. [tex]\(6x^2 - 2x + 9x - 5\)[/tex]
4. [tex]\(6x^2 - 3x + 10x - 5\)[/tex]
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