Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3x29 • x) -  30  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3x³⁰ - 30  =   3 • (x³⁰ - 10) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x³⁰ - 10 

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:

 3.3      Factoring:  x³⁰ - 10 

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  3  :

  3 • (x30 - 10)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    3   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    x³⁰-10 = 0 

 Add  10  to both sides of the equation : 
                      x³⁰ = 10
                     x  =  30th root of (10) 

 The equation has two real solutions  
 These solutions are  x = ± 30th root of 10 = ± 1.0798  
 

Two solutions were found :