Solution - Factoring binomials using the difference of squares

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     x^2+4*x-65-(4*x-1)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x²-64 

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 : 64 is the square of 8
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 8)  •  (x - 8) 

Equation at the end of step  1  :

  (x + 8) • (x - 8)  = 0 

Step  2  :

Theory - Roots of a product :

 2.1    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 :

 2.2      Solve  :    x+8 = 0 

 Subtract  8  from both sides of the equation : 
                      x = -8

Solving a Single Variable Equation :

 2.3      Solve  :    x-8 = 0 

 Add  8  to both sides of the equation : 
                      x = 8

Two solutions were found :

1.  x = 8
2.  x = -8