Question

the rectangle below has an area of $x^{2}-6x - 7$ square meters and a width of $x - 7$ meters. what expression represents the length of the rectangle? length = meters related content factoring quadratics as $(x + a)(x + b)$ factoring quadratics as $(x + a)(x + b)$ (example 2)

Explanation:

Step1: Recall the area formula for a rectangle

The area of a rectangle is \(A = \text{length}\times\text{width}\). So, \(\text{length}=\frac{A}{\text{width}}\). Given \(A=x^{2}-6x - 7\) and \(\text{width}=x - 7\), we need to perform polynomial division or factorization.

Step2: Factor the quadratic expression

Factor \(x^{2}-6x - 7\). We look for two numbers \(m\) and \(n\) such that \(m + n=-6\) and \(m\times n=-7\). The numbers are \(m = 1\) and \(n=-7\). So, \(x^{2}-6x - 7=(x + 1)(x-7)\) using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\).

Step3: Calculate the length

Since \(\text{length}=\frac{x^{2}-6x - 7}{x - 7}\) and \(x^{2}-6x - 7=(x + 1)(x-7)\), then \(\text{length}=\frac{(x + 1)(x - 7)}{x - 7}\). Canceling out the common factor \((x - 7)\) (for \(x
eq7\)), we get \(\text{length}=x + 1\).

Answer:

\(x + 1\)