Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "b1"   was replaced by   "b^1". 

Rearrange:

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

                     14*b^1-(-12*b^3)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (14 • (b1)) -  (0 -  (22•3b3))  = 0 
 Step  2  :

Equation at the end of step  2  :

  (2•7b) -  ( -22•3b3)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   12b³ + 14b  =   2b • (6b² + 7) 

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(b) = 6b² + 7
Polynomial Roots Calculator is a set of methods aimed at finding values of  b  for which   F(b)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  b  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  6  and the Trailing Constant is  7.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,6
 of the Trailing Constant :  1 ,7

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  4  :

  2b • (6b2 + 7)  = 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  :    2b = 0 

 Divide both sides of the equation by 2:
                     b = 0

Solving a Single Variable Equation :

 5.3      Solve  :    6b²+7 = 0 

 Subtract  7  from both sides of the equation : 
                      6b² = -7
Divide both sides of the equation by 6:
                     b² = -7/6 = -1.167
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      b  =  ± √ -7/6  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -7/6  =
                    √ -1• 7/6   =
                    √ -1 •√  7/6   =
                    i •  √ 7/6

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      b=  0.0000 + 1.0801 i 
                      b=  0.0000 - 1.0801 i 

Three solutions were found :

1.   b=  0.0000 - 1.0801 i 
   
2.   b=  0.0000 + 1.0801 i 
   
3.  b = 0