Question

the rectangle below has an area of $x^2 - 9$ square meters and a width of $x - 3$ meters. what expression represents the length of the rectangle? image of a rectangle with width labeled $x - 3$, area labeled $x^2 - 9$, and length labeled as length length = blank meters

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by the formula \( A = \text{length} \times \text{width} \). So, to find the length, we can rearrange this formula to \( \text{length} = \frac{A}{\text{width}} \).

Step2: Factor the area expression

The area is given as \( x^2 - 9 \). Notice that this is a difference of squares, which factors as \( a^2 - b^2 = (a + b)(a - b) \). Here, \( a = x \) and \( b = 3 \), so \( x^2 - 9 = (x + 3)(x - 3) \).

Step3: Divide the factored area by the width

The width is \( x - 3 \). So, the length is \( \frac{(x + 3)(x - 3)}{x - 3} \). We can cancel out the common factor of \( x - 3 \) (assuming \( x
eq 3 \), which is valid for the expression representing length) to get \( x + 3 \).

Answer:

\( x + 3 \)