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]
