At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's tackle the problem of factoring the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex]. To factorize this, we need to express it as a product of two binomials.
### Step-by-Step Solution:
1. Identify the quadratic expression:
We start with the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex].
2. Factor the expression:
We want to factor the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into the form [tex]\((ax + b)(cx + d)\)[/tex].
Through factoring, we find that:
[tex]\[ 2x^2 + 7x + 6 = (x + 2)(2x + 3) \][/tex]
3. Equation of Area Representation:
To represent this algebraically, we express that the area of the rectangle formed by sides [tex]\((x + 2)\)[/tex] and [tex]\((2x + 3)\)[/tex] is equivalent to the original quadratic expression.
Therefore:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
[tex]\[ 2x^2 + 7x + 6 = (x + 2)(2x + 3) \][/tex]
4. Verify by Multiplying the Factors to get the Original Expression:
Let's expand the factors to ensure we get back the original expression:
[tex]\[ (x + 2)(2x + 3) = x \cdot 2x + x \cdot 3 + 2 \cdot 2x + 2 \cdot 3 \][/tex]
[tex]\[ = 2x^2 + 3x + 4x + 6 \][/tex]
[tex]\[ = 2x^2 + 7x + 6 \][/tex]
Thus, the factors multiply to give us the original quadratic expression, confirming that our factorization is correct.
### Rectangular Sketch:
To visualize it, you can imagine the rectangle with sides [tex]\(x + 2\)[/tex] and [tex]\(2x + 3\)[/tex]:
- The length of the rectangle is [tex]\(x + 2\)[/tex].
- The width of the rectangle is [tex]\(2x + 3\)[/tex].
### Conclusion:
We have successfully factored the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into [tex]\((x + 2)(2x + 3)\)[/tex]. This means it can be represented as a rectangular area where one side is [tex]\(x + 2\)[/tex] and the other side is [tex]\(2x + 3\)[/tex]. The area of this rectangle matches the original quadratic expression when multiplied out, confirming the factorization.
Thus, [tex]\(2x^2 + 7x + 6\)[/tex] can indeed be factored, and the area equivalently represents the product of its side length factors.
### Step-by-Step Solution:
1. Identify the quadratic expression:
We start with the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex].
2. Factor the expression:
We want to factor the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into the form [tex]\((ax + b)(cx + d)\)[/tex].
Through factoring, we find that:
[tex]\[ 2x^2 + 7x + 6 = (x + 2)(2x + 3) \][/tex]
3. Equation of Area Representation:
To represent this algebraically, we express that the area of the rectangle formed by sides [tex]\((x + 2)\)[/tex] and [tex]\((2x + 3)\)[/tex] is equivalent to the original quadratic expression.
Therefore:
[tex]\[ \text{Area} = \text{Length} \times \text{Width} \][/tex]
[tex]\[ 2x^2 + 7x + 6 = (x + 2)(2x + 3) \][/tex]
4. Verify by Multiplying the Factors to get the Original Expression:
Let's expand the factors to ensure we get back the original expression:
[tex]\[ (x + 2)(2x + 3) = x \cdot 2x + x \cdot 3 + 2 \cdot 2x + 2 \cdot 3 \][/tex]
[tex]\[ = 2x^2 + 3x + 4x + 6 \][/tex]
[tex]\[ = 2x^2 + 7x + 6 \][/tex]
Thus, the factors multiply to give us the original quadratic expression, confirming that our factorization is correct.
### Rectangular Sketch:
To visualize it, you can imagine the rectangle with sides [tex]\(x + 2\)[/tex] and [tex]\(2x + 3\)[/tex]:
- The length of the rectangle is [tex]\(x + 2\)[/tex].
- The width of the rectangle is [tex]\(2x + 3\)[/tex].
### Conclusion:
We have successfully factored the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into [tex]\((x + 2)(2x + 3)\)[/tex]. This means it can be represented as a rectangular area where one side is [tex]\(x + 2\)[/tex] and the other side is [tex]\(2x + 3\)[/tex]. The area of this rectangle matches the original quadratic expression when multiplied out, confirming the factorization.
Thus, [tex]\(2x^2 + 7x + 6\)[/tex] can indeed be factored, and the area equivalently represents the product of its side length factors.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.