Question 1197474
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x = width of the uniform strip (i.e. the gap between the rug edge and the wall)


Diagram
<img width=400 src = "https://i.imgur.com/qrOvroG.png">
The larger rectangle is 19 by 31
The smaller rectangle inside is (19-2x) by (31-2x)


The smaller rectangle area is 253 sq ft
length*width = area
(19-2x)*(31-2x) = 253
19*(31-2x) - 2x*(31-2x) = 253
589 - 38x - 62x + 4x^2 = 253
589 - 100x + 4x^2 = 253
589 - 100x + 4x^2 - 253 = 0
4x^2 - 100x + 336 = 0


Compare this to ax^2+bx+c = 0
a = 4
b = -100
c = 336


Use the quadratic formula to isolate x.
{{{x = (-b+-sqrt(b^2-4ac))/(2a)}}}


{{{x = (-(-100)+-sqrt((-100)^2-4(4)(336)))/(2(4))}}}


{{{x = (100+-sqrt(10000-5376))/(8)}}}


{{{x = (100+-sqrt(4624))/(8)}}}


{{{x = (100+-  68)/(8)}}}


{{{x = (100+68)/(8)}}} or {{{x = (100-68)/(8)}}}


{{{x = (168)/(8)}}} or  {{{x = (32)/(8)}}}


{{{x = 21}}} or  {{{x = 4}}}


If x = 21, then 19-2x = 19-2*21 = 19-42 = -23 which isn't possible. 
A negative length or width makes no sense.
We'll ignore x = 21.


The only practical solution is x = 4.
If x = 4, then 19-2x = 19-2*4 = 19-8 = 11
If x = 4, then 31-2x = 31-2*4 = 31-8 = 23
Both results are positive.
The carpet is 11 by 23 to get an area of 11*23 = 253 sq ft. This verifies the answer.


Answer: <font color=red>11 ft by 23 ft</font>
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