Find an expression for the length of a side of a square with an area of 9n^2+54n+81

  s = 3n + 9 .

Explanation:

All squares are rectangles.

A square is a rectangle with 4 (FOUR) EQUAL side lengths.

A square has the same length and same length.

The formula for the area, "A", of a square is:

A = s² ;  in which "s" is the side length of the square.

So,  given:  A = 9n² + 54n + 81 ;  Find "s" ; 

→ A = s² ;

↔  s² = A ; 

Plug in our value given for "A" ; 

→ s² = 9n² + 54n+ 81 ;

Take the positive square root of EACH SIDE of the equation; to isolate "s" on one side of the equation; & to solve for "s" (the side length); 

Note:  We take the "positive" square root" ;  since the "side length of a square cannot be a "negative value" ; 

→  √(s²)  =  √(9n² + 54n + 81)  ; 

Take the value under the square root sign:

"9n² + 54n + 81" ; and factor out a "9" ;

→ 9(n² +6n + 9)

And rewrite as: 

√[9(n² + 6n + 9)] ; 

Note:  √[9(n² + 6n + 9)] = √9 * √(n² + 6n + 9) ;

 √9 = 3 ; 

√(n² + 6n + 9) is a perfect square;  that is:  √(n² + 6n + 9) = (n + 3 ) ;

So, √9 *√(n² + 6n + 9) = 3 * (n + 3) = 3*n  + 3*3 = 3n + 9 ; 

Rewrite:

√(s²) = 3n + 9 ; 

 →    s = 3n + 9 .