Solution - Nonlinear equations

Rearrange:

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

                     10*x^2-19-(5)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•5x2) -  19) -  5  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   10x² - 24  =   2 • (5x² - 12) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  5x² - 12 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  5  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Equation at the end of step  3  :

  2 • (5x2 - 12)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    5x²-12 = 0 

 Add  12  to both sides of the equation : 
                      5x² = 12
Divide both sides of the equation by 5:
                     x² = 12/5 = 2.400
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 12/5  

 The equation has two real solutions  
 These solutions are  x = ±√ 2.400 = ± 1.54919  
 

Two solutions were found :