Brainliess asap 20 point

part a: the area of a square is (9x^2 + 24x + 16) square units. determine the length of each side of the square by factoring the area expression completely. show your work. (5 points)

part b: the area of a rectangle is (16x^2 − 25y^2) square units. determine the dimensions of the rectangle by factoring the area expression completely. show your work. (5 points)

Answers:  

   Part A)  " (3x + 4) " units  . 

   Part B)  "The dimensions of the rectangle are:

                             " (4x + 5y) " units ;  AND:  " (4x − 5y)"  units."


Explanation for  Part A):


Since each side length of a square is the same; 
   
    Area = Length * width = L * w ;  L = w  = s = s ;

      in which:  " s = side length" ;

So, the Area of a square, "A"  = L * w = s * s = s² ;

{Note:  A "square" is a rectangle with 4 (four) equal sides.}.

→  Each side length, "s", of a square is equal.

Given:  s² = "(9x² + 24x + 16)" square units ;

Find "s" by factoring: "(9x² + 24x + 16)" completely:

   →  " 9x² + 24x + 16 ";

Factor by "breaking into groups" :

"(9x² + 24x + 16)"  = 

    →  "(9x² + 12x) (12x + 16)" ;


Given:   " (9x² + 24x + 16) " ; 

Let us start with the term:


" (9x² + 12x) " ; 

    →  Factor out a "3x" ;  → as follows:


    → " 3x (3x + 4) " ; 

Then, take the term:

    → " (12x + 16) " ;

And factor out a "4" ;   →  as follows:


    → " 4 (3x + 4) " 

We have:

" 9x² + 24x + 16 " ;

    =  " 3x (3x + 4)  +  4(3x + 4) " ;

Now, notice the term:  "(3x + 4)" ; 

We can "factor out" this term:

3x (3x + 4)  +  4(3x + 4)  = 

     →  " (3x + 4) (3x + 4) " .  → which is the fully factored form of:

                                                   " 9x² + 24x + 16 "  ; 

     →  Or; write:  "  (3x + 4) (3x + 4)" ; as:  " (3x + 4)² " .

     →  So,  "s² = 9x² + 24x + 16 " ; 

Rewrite as:  " s² = (3x + 4)² " .

     →  Solve for the "positive value of "s" ; 

     →  {since the "side length of a square" cannot be a "negative" value.}.

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

     →  ⁺√(s²)  =  ⁺√[(3x + 4)²]   '

To get:

     →  s  = " (3x + 4)" units .


Part A):  The answer is:  "(3x + 4)" units.


Explanation for Part B):


The area, "A" of a rectangle is:

    A = L * w ;  

 in which "A" is the "area" of the rectangle;
                "L" is the "length" of the rectangle; 
                "w" is the "width" of the rectangle; 

  Given:  " A = (16x² − 25y²) square units" ;  
   
       →  We are asked to find the dimensions, "L" & "w" ;
       →  by factoring the given "area" expression completely:

  → Factor:  " (16x² − 25y²) square units " completely '

Note that:  "16" and: "25" are both "perfect squares" ;
      
We can rewrite: " (16x² − 25y²) "  ;   as:

       =   " (4²x²)  −  (5²y²) " ; and further rewrite the expression:

Note:  

" (16x²) " ;  can be written as:  "(4x)² " ;

 ↔ " (4x)²  =  "(4²)(x²)" = 16x² "

Note:  The following property of exponents:

         →  (xy)ⁿ = xⁿ yⁿ ;    →  As such:  " 16x² = (4²x²) = (4x)² " . 

Note:


     →   " (25x²) " ;  can be written as:  " (5x)² " ; 

        ↔   "( 5x)²  =  "(5²)(x²)" = 25x² " ; 

Note:  The following property of exponents:

         →  (xy)ⁿ = xⁿ yⁿ ;    →  As such:  " 25x² = (5²x²) = (5x)² " . 


→  So, we can rewrite:  " (16x² − 25y²) " ;  

as:  " (4x)² − (5y)² " ;   
 
    → {Note:  We substitute: "(4x)² "  for "(16x²)" ; & "(5y)² "  for "(25y²)" .} . ; 

→  We have:  " (4x)² − (5y)² " ;

→  Note that we are asked to "factor completely" ; 

→  Note that:  " x² − y² = (x + y) (x − y) " ;

      → {This property is known as the "difference of squares".}.

→ As such:  " (4x)² − (5y)² " = " (4x + 5y) (4x − 5y) " .

Part B):  The answer is:  "The dimensions of the rectangle are:

                              " (4x + 5y) " units ;  AND:  " (4x − 5y)"  units."