Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (k4) -  (22•52k2)  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   k⁴ - 100k²  =   k² • (k² - 100) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  k² - 100 

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 :  k²  is the square of  k¹ 

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

Equation at the end of step  3  :

  k2 • (k + 10) • (k - 10)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    k² = 0 

  Solution is  k² = 0

Solving a Single Variable Equation :

 4.3      Solve  :    k+10 = 0 

 Subtract  10  from both sides of the equation : 
                      k = -10

Solving a Single Variable Equation :

 4.4      Solve  :    k-10 = 0 

 Add  10  to both sides of the equation : 
                      k = 10

Three solutions were found :

1.  k = 10
2.  k = -10
3.  k² = 0