Solution - Linear equations with one unknown

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "k1"   was replaced by   "k^1". 

Rearrange:

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

                     -k^4*(k^1)-(2*k)=0 

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   -k⁵ - 2k  =   -k • (k⁴ + 2) 

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(k) = k⁴ + 2
Polynomial Roots Calculator is a set of methods aimed at finding values of  k  for which   F(k)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  k  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  1  and the Trailing Constant is  2.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

  -k • (k4 + 2)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    -k = 0 

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

Solving a Single Variable Equation :

 3.3      Solve  :    k⁴+2 = 0 

 Subtract  2  from both sides of the equation : 
                      k⁴ = -2
                     k  =  ∜ -2  

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

                      k=  0.8409 + 0.8409 i 
                      k=  -0.8409 + 0.8409 i 
                      k=  -0.8409 - 0.8409 i 
                      k=  0.8409 - 0.8409 i 

5 solutions were found :

1.   k=  0.8409 - 0.8409 i 
   
2.   k=  -0.8409 - 0.8409 i 
   
3.   k=  -0.8409 + 0.8409 i 
   
4.   k=  0.8409 + 0.8409 i 
   
5.  k = 0