Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  34x2 -  625  = 0 

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  81x²-625 

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 :  81  is the square of  9 
Check : 625 is the square of 25
Check :  x²  is the square of  x¹ 

Factorization is :       (9x + 25)  •  (9x - 25) 

Equation at the end of step  2  :

  (9x + 25) • (9x - 25)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 3.2      Solve  :    9x+25 = 0 

 Subtract  25  from both sides of the equation : 
                      9x = -25
Divide both sides of the equation by 9:
                     x = -25/9 = -2.778

Solving a Single Variable Equation :

 3.3      Solve  :    9x-25 = 0 

 Add  25  to both sides of the equation : 
                      9x = 25
Divide both sides of the equation by 9:
                     x = 25/9 = 2.778

Two solutions were found :

1.  x = 25/9 = 2.778
   
2.  x = -25/9 = -2.778