Question

the rectangle below has an area of $x^{2}-15x + 56$ square meters and a length of $x - 7$ meters.
what expression represents the width of the rectangle?
$x - 7$
width
$x^{2}-15x + 56$
width $=square$ meters

Explanation:

Step1: Recall the formula for the area of a rectangle

The area of a rectangle \(A = \text{length}\times\text{width}\). Given \(A=x^{2}-15x + 56\) and \(\text{length}=x - 7\), then \(\text{width}=\frac{A}{\text{length}}=\frac{x^{2}-15x + 56}{x - 7}\).

Step2: Factor the quadratic expression in the numerator

We factor \(x^{2}-15x + 56\). We need two numbers \(a\) and \(b\) such that \(a + b=-15\) and \(a\times b = 56\). The numbers are \(-7\) and \(-8\) since \((-7)+(-8)=-15\) and \((-7)\times(-8) = 56\). So \(x^{2}-15x + 56=(x - 7)(x - 8)\).

Step3: Simplify the rational expression

Substitute the factored form into the width formula: \(\text{width}=\frac{(x - 7)(x - 8)}{x - 7}\). Cancel out the common factor \((x - 7)\) (assuming \(x
eq7\)).

Answer:

\(x - 8\)