The area of a rectangle is 2 square feet. the perimeter of the rectangle is 8 ft. Find the length,l, and width, w, of the rectangle. What is l^2 and w^2?

If the area of a rectangle is 2 feet, and its perimeter is 8 feet, we can have two equations as follows:
Area(a)= length(l) * width(w)
Perimeter(p) = 2*length(l)+2*width(w)
Because we have two variables (l and w) and two equations, we can use one equation to solve for l in terms of w, and then plug in that equation to the second equation, in order that we end up with one equation and one variable, as follows:
A=l*w =2
P=2l +2w = 8; l=(8-2w)/2
Thus, l*w=((8-2w)/2)*w = 2
solving using quadratic equation, we get: -2w^2+8w-4 =0.
w =.586, and w = 3.414.
Thus, if width (w) = .586, then length (l) = 3.414.
It follows that w^2 = .343396, and l^2=11.655396.

Given data:
Area of rectangle=2 sq feet.
perimeter = 8 ft.
length = l breadth = b
now area = l x b=2
perimeter =2(l+b)=8
now l + b = 4
l=b-4--- substitute in area eqn
l x b=2
(b-4)xb=2
b^2 -4b-2=0
b=4.5 ft or b=0.5 ft
so length = 0.5 ft or 4.5 ft.
l^2 = 4.5 x 4.5=20.25
b^2 = 0.5 x 0.5 = 0.25

Because both the area and perimeter are related to the length and width, this can be represented as a system of equations. If the area of a rectangle is 2 feet, and its perimeter is 8 feet, we want to use the perimeter and area equations, as follows:
Area(a)= length(l) * width(w)
Perimeter(p) = 2*length(l)+2*width(w)
Because we have two variables (l and w) and two equations, we can use one equation to solve for l in terms of w, and then plug in that equation to the second equation, in order that we end up with one equation and one variable, as follows:

A=l*w =2
P=2l +2w = 8; l=(8-2w)/2
Thus, l*w=((8-2w)/2)*w = 2
We can re-write this equation in standard form quadratic:
-2w^2+8w-4 =0. This yields solutions (using the "quadratic formula," explained at the link below), w =.586, and w = 3.414. Whichever answer we select for the width (w) w will determine that the other answer is the length (l). Thus, if width (w) = .586, then length (l) = 3.414. It follows that w^2 = .343396, and l^2=11.655396.
https://www.purplemath.com/modules/quadform.htm