Solution - Nonlinear equations

Rearrange:

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

                     x^2-(75)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x²-75 

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 : 75 is not a square !!

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

Equation at the end of step  1  :

  x2 - 75  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    x²-75 = 0 

 Add  75  to both sides of the equation : 
                      x² = 75
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 75  

 Can  √ 75 be simplified ?

Yes!   The prime factorization of  75   is
   3•5•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).

√ 75   =  √ 3•5•5   =
                ±  5 • √ 3

The equation has two real solutions  
 These solutions are  x = 5 • ± √3 = ± 8.6603  
 

Two solutions were found :