Question
A rectangle has an area of x^2-8x-65 Which expression could represent the width of the rectangle?
Answers
GPT-4o mini
The area of a rectangle is given by the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
In your case, the area is given as \( x^2 - 8x - 65 \). To find an expression for the width, we can factor the quadratic expression.
We want to factor \( x^2 - 8x - 65 \). To do this, we look for two numbers that multiply to \(-65\) (the constant term) and add to \(-8\) (the coefficient of the \(x\) term).
The numbers \(-13\) and \(5\) fit this requirement, since:
\[
-13 \times 5 = -65 \quad \text{and} \quad -13 + 5 = -8
\]
Thus, we can factor the quadratic expression as follows:
\[
x^2 - 8x - 65 = (x - 13)(x + 5)
\]
This shows that the area of the rectangle can be expressed as the product of the factors \( (x - 13) \) and \( (x + 5) \).
If we let one factor represent the length and the other factor represent the width, either \( (x - 13) \) or \( (x + 5) \) could serve as the width of the rectangle.
Thus, possible expressions for the width of the rectangle could be:
\[
\boxed{x - 13} \quad \text{or} \quad \boxed{x + 5}
\]
\[
\text{Area} = \text{length} \times \text{width}
\]
In your case, the area is given as \( x^2 - 8x - 65 \). To find an expression for the width, we can factor the quadratic expression.
We want to factor \( x^2 - 8x - 65 \). To do this, we look for two numbers that multiply to \(-65\) (the constant term) and add to \(-8\) (the coefficient of the \(x\) term).
The numbers \(-13\) and \(5\) fit this requirement, since:
\[
-13 \times 5 = -65 \quad \text{and} \quad -13 + 5 = -8
\]
Thus, we can factor the quadratic expression as follows:
\[
x^2 - 8x - 65 = (x - 13)(x + 5)
\]
This shows that the area of the rectangle can be expressed as the product of the factors \( (x - 13) \) and \( (x + 5) \).
If we let one factor represent the length and the other factor represent the width, either \( (x - 13) \) or \( (x + 5) \) could serve as the width of the rectangle.
Thus, possible expressions for the width of the rectangle could be:
\[
\boxed{x - 13} \quad \text{or} \quad \boxed{x + 5}
\]