The area of a rooftop can be expressed as 9x^2+6x+1. the rooftop is a quadrilateral. what type of quadrilateral is the rooftop? justify your answer. if the area of the rooftop is 361m^2 , what is the length of one side of the rooftop

The kind of quadrilateral that characterizes the roof top is a Square: and the length of one side of the Square is; 19m

By factorisation; the area of the roof top can be factorised as follows;

Area = 9x² + 6x + 1 = (3x + 1) (3x + 1)

Since, the area is said to be a quadrilateral and upon factorization; it's sides are equal;

We can conclude that the roof top is a Square.

Additionally, if the area is 361m².

The length, (3x +1 ) of one side of the rooftop is;

(3x + 1) = √361(3x + 1) = 19

Length of one side is therefore; 19m

Read more:

link

Can 9x^2+6x+1 be factored?  If so, we could then hypothesize that the resultant factors, multiplied together, quantize the area of a rooftop.

9x^2 + 6x + 1 = (3x + 1)(3x + 1).  So it looks like the length times the width of the quadrilateral is (3x + 1)^2, and because L = W, the quadrilateral is a square.

If the formula for the area is (3x + 1)^2 and the numeric value of that area is 361 m^2, then 

sqrt[ (3x + 1)^2 ] = plus or minus sqrt [361]:

3x + 1 = plus or minus 19.  Then 3x = -1 plus or minus 19, 

which produces two results:  3x = -1 + 19    and    3x = -1 - 19.

The roots are x = 20/3 m and x = -20/3 m.  A negative length makes no sense, so we choose x = 20/3 m; then y is also 20/3 m.

The length of one side of the rooftop is 20/3 m.