Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  100 -  112k2  = 0 

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  100-121k² 

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 :  100  is the square of  10 
Check : 121 is the square of 11
Check :  k²  is the square of  k¹ 

Factorization is :       (10 + 11k)  •  (10 - 11k) 

Equation at the end of step  2  :

  (11k + 10) • (10 - 11k)  = 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  :    11k+10 = 0 

 Subtract  10  from both sides of the equation : 
                      11k = -10
Divide both sides of the equation by 11:
                     k = -10/11 = -0.909

Solving a Single Variable Equation :

 3.3      Solve  :    -11k+10 = 0 

 Subtract  10  from both sides of the equation : 
                      -11k = -10
Multiply both sides of the equation by (-1) :  11k = 10


Divide both sides of the equation by 11:
                     k = 10/11 = 0.909

Two solutions were found :

1.  k = 10/11 = 0.909
   
2.  k = -10/11 = -0.909