The rectangle below has an area of \( x^{2}-6 x-7 \) square meters and width of \( x-7 \) meters. What expression represents the length of the rectangle?

To find the length of the rectangle, we can use the formula for the area of a rectangle, which is given by the area equal to width multiplied by length. Here, we have the area as \( x^{2} - 6x - 7 \) square meters and the width as \( x - 7 \) meters. We can set up the equation: \[ \text{Length} \times (x - 7) = x^{2} - 6x - 7 \] Now, to find the length, we can divide the area by the width: \[ \text{Length} = \frac{x^{2} - 6x - 7}{x - 7} \] Next, we can factor the area expression. The quadratic \( x^{2} - 6x - 7 \) factors into \( (x - 7)(x + 1) \). Thus, we simplify: \[ \text{Length} = \frac{(x - 7)(x + 1)}{(x - 7)} \] Assuming \( x \neq 7 \), we can cancel \( (x - 7) \) from the numerator and denominator, yielding: \[ \text{Length} = x + 1 \] So, the expression that represents the length of the rectangle is \( x + 1 \) meters. Now onto something fun: Did you know that rectangles have been intriguing mathematicians for centuries? The concept of area, perimeter, and the relationships within a rectangle were even studied by ancient civilizations, like the Egyptians, who used simple geometry to build their pyramids! For a practical application, understanding how to calculate the length of a rectangle is super useful in everyday tasks! Whether you're arranging furniture, planning a garden, or designing layouts for a room, knowing how to manipulate these dimensions can help you optimize space effectively!