Solution - Linear equations with one unknown

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x4"   was replaced by   "x^4".  1 more similar replacement(s).

Rearrange:

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

                     9*(x^1)-(2*x^4)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (9 • (x1)) -  2x4  = 0 
 Step  2  :

Equation at the end of step  2  :

  32x -  2x4  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   9x - 2x⁴  =   -x • (2x³ - 9) 

Trying to factor as a Difference of Cubes:

 4.2      Factoring:  2x³ - 9 

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 :  2  is not a cube !!

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

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(x) = 2x³ - 9
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  2  and the Trailing Constant is  -9.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  4  :

  -x • (2x3 - 9)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    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 :

 5.2      Solve  :    -x = 0 

 Multiply both sides of the equation by (-1) :  x = 0

Solving a Single Variable Equation :

 5.3      Solve  :    2x³-9 = 0 

 Add  9  to both sides of the equation : 
                      2x³ = 9
Divide both sides of the equation by 2:
                     x³ = 9/2 = 4.500
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 9/2  

 The equation has one real solution
This solution is  x = ∛ 4.500 = 1.65096

Two solutions were found :

1.  x = ∛ 4.500 = 1.65096
2.  x = 0