Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  5m2 +  10  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5m² + 10  =   5 • (m² + 2) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(m) = m² + 2
Polynomial Roots Calculator is a set of methods aimed at finding values of  m  for which   F(m)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  m  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  3  :

  5 • (m2 + 2)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    5   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    m²+2 = 0 

 Subtract  2  from both sides of the equation : 
                      m² = -2
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      m  =  ± √ -2  

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

Accordingly,  √ -2  =
                    √ -1• 2   =
                    √ -1 •√  2   =
                    i •  √ 2

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

                      m=  0.0000 + 1.4142 i 
                      m=  0.0000 - 1.4142 i 

Two solutions were found :

1.   m=  0.0000 - 1.4142 i 
   
2.   m=  0.0000 + 1.4142 i