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

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.