Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  23u2 -  242  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   8u² - 242  =   2 • (4u² - 121) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  4u² - 121 

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 : 121 is the square of 11
Check :  u²  is the square of  u¹ 

Factorization is :       (2u + 11)  •  (2u - 11) 

Equation at the end of step  3  :

  2 • (2u + 11) • (2u - 11)  = 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.

Equations which are never true :

 4.2      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.3      Solve  :    2u+11 = 0 

 Subtract  11  from both sides of the equation : 
                      2u = -11
Divide both sides of the equation by 2:
                     u = -11/2 = -5.500

Solving a Single Variable Equation :

 4.4      Solve  :    2u-11 = 0 

 Add  11  to both sides of the equation : 
                      2u = 11
Divide both sides of the equation by 2:
                     u = 11/2 = 5.500

Two solutions were found :

1.  u = 11/2 = 5.500
   
2.  u = -11/2 = -5.500