Question

Consider a rectangle having one side of length \(x-6\) and having an area given by \(A=x^{2}-17 x+66\) Use factoring to find an expression for the other side of the rectangle.

Short answer

The expression for the other side of the rectangle is \(x-11\).

Step-by-step solution

Factorize the Area Equation

To factorize the quadratic equation \(x^{2}-17 x+66\), you need to find numbers that multiply to 66 and add to -17. Those numbers are -11 and -6. Thus, the factorization is \((x-11)(x-6)\).

Write down the formula for finding the other side length

The formula to find the other side of the rectangle is \(A/(x-6)\) (As the area of the rectangle is calculated by multiplying length and width, we divide the total area by the given side length to find the other side length). Substitute \(A\) with the factored form generated in step 1, resulting in: \[((x-11)(x-6))/(x-6)\].

Simplification

In the above expression, \(x-6\) will cancel out each other, hence, the expression for the other side of the rectangle is \(x-11\).