The area of a rectangle is 252 square feet. If the perimeter is 66 feet, find the length and width of a rectangle. What is l^2 and w^2?

Step 1:
Given quantities:
Area of the rectangle (A) = Length (l) * Width (w) = 252 sq. feet
Perimeter of the rectangle (P) = 2 * (Length (l) + Width (w)) = 66 feet

To solve equations involving two variables, we need to have two such equations. For three variables, we need three equations and so on.
We could express the area and perimeter symbolically in terms of length and width as,
l * w = 252....... Equation (1)
2 * (l + w) = 66...... Equation (2)
Therefore,
l + w = 33, which could be expressed as, w = 33 - l

Substituting the expression 'w = 33 - l' in the equation 'l * w = 252', we get,
l * (33 - l) = 252
33l - l^2 - 252 = 0
l^2 - 33l + 252 = 0 ...... Multiplying the Left hand side and Right side of the equation by (-1)
The next step is to factorize the equation, which is as follows,
l^2 - 21l - 12l + 252 = 0
l * (l - 21) - 12 * (l - 21) = 0
(l - 21) (l - 12) = 0 ......... Equation (3)

We have obtained two solutions for length (l) = 21 or 12 feet
Plugging the values of 'l' in the equation 'w = 33 - l', we get,
Width (w) = 12 or 21 feet

Also, the values for l^2 are (21)^2 = 441 or (12)^2 = 144
Likewise, the values for w^2 are (12)^2 = 144 or (21)^2 = 441

P.S: The units are really important in the beginning as well as the concluding step.
Also, curious about how the rectangles turn out in each case. Please refer the image in the attachment

First, identify the geometric equations at hand.
First, Area: A = L * W, where L represents length and W represents width.
Perimeter: P = 2*L+ 2*W
Thus, in this example, L * W = A = 256,
and 2L + 2W = 66.

As a general rule, we know that we can solve for as many equations as we have variables. Because here we have two variables and two equations, we can simply solve for one in terms of the other using one equation, and plug that solution into the other equation, in order to eliminate one variable, as follows:
Using the perimeter equation, we can rearrange to solve for L: L = (66-2W)/2
We can plug this into the area equation: ((66-2W)/2) * W = 66
This second equation we can reconfigure as a quadratic equation in standard form, as follows:
-2W² + 66W - 504 = 0
Using the quadratic formula or a graphing utility, we have W = 12, 21. Whichever is not the width is going to be the length. Thus, if we let W = 12, then L = 21. It follows that w² =144 and l²=441.