Question

the rectangle below has an area of $x^{2}-x - 72$ square meters and a length of $x + 8$ meters. what expression represents the width of the rectangle? width = 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, \(\text{width}=\frac{A}{\text{length}}\). Here, \(A=x^{2}-x - 72\) and \(\text{length}=x + 8\). Then \(\text{width}=\frac{x^{2}-x - 72}{x + 8}\).

Step2: Factor the quadratic expression

Factor \(x^{2}-x - 72\). We need two numbers \(m\) and \(n\) such that \(m\times n=-72\) and \(m + n=-1\). The numbers are \(m = 8\) and \(n=-9\) since \(8\times(-9)=-72\) and \(8+(-9)=-1\). So, \(x^{2}-x - 72=(x + 8)(x-9)\).

Step3: Simplify the fraction

Substitute the factored form into the width formula: \(\text{width}=\frac{(x + 8)(x-9)}{x + 8}\). Cancel out the common factor \((x + 8)\) (assuming \(x
eq - 8\)).

Answer:

\(x-9\)