Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

Step  3  :

               4x2 - 1  
 Simplify   ————————————
            4x2 - 4x - 3

Trying to factor as a Difference of Squares :

 3.1      Factoring:  4x² - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  4  is the square of  2 
Check : 1 is the square of 1
Check :  x²  is the square of  x¹ 

Factorization is :       (2x + 1)  •  (2x - 1) 

Trying to factor by splitting the middle term

 3.2     Factoring  4x² - 4x - 3 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  -4x  its coefficient is  -4 .
The last term, "the constant", is  -3 

Step-1 : Multiply the coefficient of the first term by the constant   4 • -3 = -12 

Step-2 : Find two factors of  -12  whose sum equals the coefficient of the middle term, which is   -4 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  2 
                     4x² - 6x + 2x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2x • (2x-3)
              Add up the last 2 terms, pulling out common factors :
                     1 • (2x-3)
Step-5 : Add up the four terms of step 4 :
                    (2x+1)  •  (2x-3)
             Which is the desired factorization

Canceling Out :

 3.3    Cancel out  (2x+1)  which appears on both sides of the fraction line.

Final result :

  2x - 1
  ——————
  2x - 3