Solution - Linear equations with one unknown

Rearrange:

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

                     13-(32*(7*x^3))=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  13 -  (32 • 7x3)  = 0 

Step  2  :

Equation at the end of step  2  :

  13 -  (25•7x3)  = 0 

Step  3  :

Trying to factor as a Difference of Cubes:

 3.1      Factoring:  13-224x³ 

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 :  13  is not a cube !!

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

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = -224x³+13
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  13  and the Trailing Constant is  -224.

 The factor(s) are:

of the Leading Coefficient :  1,13
 of the Trailing Constant :  1 ,2 ,4 ,7 ,8 ,14 ,16 ,28 ,32 ,56 , etc

 Let us test ....

Note - For tidiness, printing of 35 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  13 - 224x3  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    -224x³+13 = 0 

 Subtract  13  from both sides of the equation : 
                      -224x³ = -13
Multiply both sides of the equation by (-1) :  224x³ = 13


Divide both sides of the equation by 224:
                     x³ = 13/224 = 0.058
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 13/224  

 The equation has one real solution
This solution is  x = ∛ 0.058 = 0.38717

One solution was found :