Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x6"   was replaced by   "x^6". 

Rearrange:

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

                     0-(-x^2-5*x^6)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  0 -  ((0 -  (x2)) -  5x6)  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5x⁶ + x²  =   x² • (5x⁴ + 1) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = 5x⁴ + 1
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  5  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  x2 • (5x4 + 1)  = 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  :    x² = 0 

  Solution is  x² = 0

Solving a Single Variable Equation :

 4.3      Solve  :    5x⁴+1 = 0 

 Subtract  1  from both sides of the equation : 
                      5x⁴ = -1
Divide both sides of the equation by 5:
                     x⁴ = -1/5 = -0.200
                     x  =  ∜ -1/5  

 The equation has no real solutions. It has 4 imaginary, or complex solutions.

                      x=  0.4729 + 0.4729 i 
                      x=  -0.4729 + 0.4729 i 
                      x=  -0.4729 - 0.4729 i 
                      x=  0.4729 - 0.4729 i 

5 solutions were found :

1.   x=  0.4729 - 0.4729 i 
   
2.   x=  -0.4729 - 0.4729 i 
   
3.   x=  -0.4729 + 0.4729 i 
   
4.   x=  0.4729 + 0.4729 i 
   
5.  x² = 0