What is a permeter for a dining room?

Here's what we know:
A = Lw
A = 24 (Area is length times with, and it equals 24 m2)

P = 2L + 2w
P = 22 (Perimeter is the 2 lengths and 2 widths added together, and that's 22 m)

22 = 2L + 2w
Factor out a 2:
11 = L + w
Isolate a variable:
11 - w = L

Now to use that to work with the area:
24 = w(11 - w)
Distribute:
24 = -w2 + 11w
Move everything to one side to make a quadratic (add w2, subtract 11w)
w2 - 11w + 24 = 0

With the b-value being negative (-11) and c positive (+24), when we factor out the quadratic, we know that they're both going to use a subtraction sign:
(w - m)(w - n) = 0
We now need two numbers that when "added" together make -11, and multiplied to make 24:
-5, -6 → 11, 30
-4, -7 → 11, 28
-3, -8 → 11, 24
(w - 3)(w - 8) = 0
0 = w - 3, or 0 = w - 8
w = 3 meters and 8 meters for the dimensions

Check the answer with the perimeter:
2 * 3 + 2 * 8 = 22
6 + 16 = 22
22 = 22

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