No solutions found

Rearrange:

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

                     18*x-(-16*x^31)=0 

Step  1  :

Equation at the end of step  1  :

  18x -  (0 -  24x31)  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   16x³¹ + 18x  =   2x • (8x³⁰ + 9) 

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  8x³⁰ + 9 

Theory : A sum 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 :  8  is the cube of  2 

Check :  9  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  3  :

  2x • (8x30 + 9)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    2x = 0 

 Divide both sides of the equation by 2:
                     x = 0

Solving a Single Variable Equation :

 4.3      Solve  :    8x³⁰+9 = 0 

 Subtract  9  from both sides of the equation : 
                      8x³⁰ = -9
Divide both sides of the equation by 8:
                     x³⁰ = -9/8 = -1.125
                     x  =  30th root of (-9/8) 

 The equation has no real solutions. It has 30 imaginary, or complex solutions.
 These solutions are x = 30th root of -1.12500

1.  These solutions are x = 30th root of -1.12500
2.  x = 0