Final answer:
To find the dimensions of Tammy's rug including its border, an equation is set up based on the total area after the border is added. The width of the rug is represented as w and the length is w+4, leading to the equation (w+8)(w+4)=77 or w^2+12w-45=0, which is a quadratic equation.
Step-by-step explanation:
To determine the length and width of Tammy's rug including the border, we must first let w represent the width of the rug. Then, the length will be w + 4 since the rug is 4 feet longer than its width. Including the 2-foot border around the rug, the overall dimensions will be w + 4 (for the length) plus 4 more feet (2 feet on each side) and w plus 4 feet (2 feet for each border) for the width.
The equation representing the total area of the rug including its border, which is 77 square feet, can be set up as follows:
(w + 4 + 4)(w + 2 + 2) = 77
This simplifies to:
(w + 8)(w + 4) = 77
When we expand this equation, we get a quadratic equation:
w^2 + 8w + 4w + 32 = 77
Simplifying further:
w^2 + 12w +32 - 77 = 0
Therefore, the quadratic equation to determine the width of the rug is:
w^2 + 12w - 45 = 0