Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22x25 • x) -  9  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  4x²⁶-9 

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 : 9 is the square of 3
Check :  x²⁶  is the square of  x¹³ 

Factorization is :       (2x¹³ + 3)  •  (2x¹³ - 3) 

Equation at the end of step  2  :

  (2x13 + 3) • (2x13 - 3)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    2x¹³+3 = 0 

 Subtract  3  from both sides of the equation : 
                      2x¹³ = -3
Divide both sides of the equation by 2:
                     x¹³ = -3/2 = -1.500
                     x  =  13th root of (-3/2) 

 Negative numbers have real 13th roots.
 13th root of (-3/2) = ¹³√ -1• 3/2  = ¹³√ -1 • ¹³√ 3/2  =(-1)•¹³√ 3/2 

The equation has one real solution, a negative number This solution is  x = 13th root of (-1.500) = -1.03168

Solving a Single Variable Equation :

 3.3      Solve  :    2x¹³-3 = 0 

 Add  3  to both sides of the equation : 
                      2x¹³ = 3
Divide both sides of the equation by 2:
                     x¹³ = 3/2 = 1.500
                     x  =  13th root of (3/2) 

 The equation has one real solution
This solution is  x = 13th root of ( 1.500) = 1.03168

Two solutions were found :

1.  x = 13th root of ( 1.500) = 1.03168
2.  x = 13th root of (-1.500) = -1.03168