Solution - Nonlinear equations

Rearrange:

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

                     4*x^2-5-(75)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22x2 -  5) -  75  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   4x² - 80  =   4 • (x² - 20) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x² - 20 

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 : 20 is not a square !!

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

Equation at the end of step  3  :

  4 • (x2 - 20)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    4   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    x²-20 = 0 

 Add  20  to both sides of the equation : 
                      x² = 20
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 20  

 Can  √ 20 be simplified ?

Yes!   The prime factorization of  20   is
   2•2•5 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 20   =  √ 2•2•5   =
                ±  2 • √ 5

The equation has two real solutions  
 These solutions are  x = 2 • ± √5 = ± 4.4721  
 

Two solutions were found :