Question
A farmer is planting his crops, and he knows what the total area of land is x^2 + 12x + 27. Based on this information, what are the width and height?
Answers
Answered by
GPT-4o mini
To find the width and height of the land that has an area represented by the expression \(x^2 + 12x + 27\), we need to factor this quadratic expression.
First, we can rewrite \(x^2 + 12x + 27\) as a product of two binomials. Specifically, we need to find two numbers that multiply to 27 (the constant term) and add to 12 (the coefficient of the linear term).
The two numbers that satisfy these conditions are 3 and 9, since:
- \(3 \times 9 = 27\)
- \(3 + 9 = 12\)
Thus, we can factor the expression as follows:
\[
x^2 + 12x + 27 = (x + 3)(x + 9)
\]
In this factorization, \(x + 3\) can be considered as the width and \(x + 9\) as the height, or vice versa. Therefore, the width and height of the planted area can be expressed as:
- Width: \(x + 3\)
- Height: \(x + 9\)
So the width and height of the area are \(x + 3\) and \(x + 9\) respectively.
First, we can rewrite \(x^2 + 12x + 27\) as a product of two binomials. Specifically, we need to find two numbers that multiply to 27 (the constant term) and add to 12 (the coefficient of the linear term).
The two numbers that satisfy these conditions are 3 and 9, since:
- \(3 \times 9 = 27\)
- \(3 + 9 = 12\)
Thus, we can factor the expression as follows:
\[
x^2 + 12x + 27 = (x + 3)(x + 9)
\]
In this factorization, \(x + 3\) can be considered as the width and \(x + 9\) as the height, or vice versa. Therefore, the width and height of the planted area can be expressed as:
- Width: \(x + 3\)
- Height: \(x + 9\)
So the width and height of the area are \(x + 3\) and \(x + 9\) respectively.