Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "n2"   was replaced by   "n^2". 

Rearrange:

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

                     5*n^2-20*n^25-(n^2)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((5 • (n2)) -  (22•5n25)) -  n2  = 0 
 Step  2  :

Equation at the end of step  2  :

  (5n2 -  (22•5n25)) -  n2  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4n² - 20n²⁵  =   -4n² • (5n²³ - 1) 

Equation at the end of step  4  :

  -4n2 • (5n23 - 1)  = 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  :    -4n² = 0 

 Multiply both sides of the equation by (-1) :  4n² = 0


Divide both sides of the equation by 4:
                     n² = 0
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      n  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  n = 0

Solving a Single Variable Equation :

 5.3      Solve  :    5n²³-1 = 0 

 Add  1  to both sides of the equation : 
                      5n²³ = 1
Divide both sides of the equation by 5:
                     n²³ = 1/5 = 0.200
                     n  =  23rd root of (1/5) 

 The equation has one real solution
This solution is  n = 23rd root of ( 0.200) = 0.93242

Two solutions were found :

1.  n = 23rd root of ( 0.200) = 0.93242
2.  n = 0