Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x²⁶-36 

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 : 36 is the square of 6
Check :  x²⁶  is the square of  x¹³ 

Factorization is :       (x¹³ + 6)  •  (x¹³ - 6) 

Equation at the end of step  1  :

  (x13 + 6) • (x13 - 6)  = 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  :    x¹³+6 = 0 

 Subtract  6  from both sides of the equation : 
                      x¹³ = -6
                     x  =  13th root of (-6) 

 Negative numbers have real 13th roots.
 13th root of (-6) = ¹³√ -1• 6  = ¹³√ -1 • ¹³√ 6  =(-1)•¹³√ 6 

The equation has one real solution, a negative number This solution is  x = negative 13th root of 6 = -1.1478

Solving a Single Variable Equation :

 2.3      Solve  :    x¹³-6 = 0 

 Add  6  to both sides of the equation : 
                      x¹³ = 6
                     x  =  13th root of (6) 

 The equation has one real solution
This solution is  x = 13th root of 6 = 1.1478

Two solutions were found :

1.  x = 13th root of 6 = 1.1478
2.  x = negative 13th root of 6 = -1.1478