Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•5k29) • k) -  1  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  10k³⁰-1 

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 :  10  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor as a Difference of Cubes:

 2.2      Factoring:  10k³⁰-1 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  10  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  2  :

  10k30 - 1  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    10k³⁰-1 = 0 

 Add  1  to both sides of the equation : 
                      10k³⁰ = 1
Divide both sides of the equation by 10:
                     k³⁰ = 1/10 = 0.100
                     k  =  30th root of (1/10) 

 The equation has two real solutions  
 These solutions are  k = 30th root of ( 0.100) = ± 0.92612  
 

Two solutions were found :