Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2•32x2) -  98  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   18x² - 98  =   2 • (9x² - 49) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  9x² - 49 

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 :  9  is the square of  3 
Check : 49 is the square of 7
Check :  x²  is the square of  x¹ 

Factorization is :       (3x + 7)  •  (3x - 7) 

Equation at the end of step  3  :

  2 • (3x + 7) • (3x - 7)  = 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  :    3x+7 = 0 

 Subtract  7  from both sides of the equation : 
                      3x = -7
Divide both sides of the equation by 3:
                     x = -7/3 = -2.333

Solving a Single Variable Equation :

 4.4      Solve  :    3x-7 = 0 

 Add  7  to both sides of the equation : 
                      3x = 7
Divide both sides of the equation by 3:
                     x = 7/3 = 2.333

Two solutions were found :

1.  x = 7/3 = 2.333
   
2.  x = -7/3 = -2.333