Solution - Nonlinear equations

Rearrange:

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

                     z^2-32-(-25)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  z²-7 

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 : 7 is not a square !!

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

Equation at the end of step  1  :

  z2 - 7  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    z²-7 = 0 

 Add  7  to both sides of the equation : 
                      z² = 7
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      z  =  ± √ 7  

 The equation has two real solutions  
 These solutions are  z = ± √7 = ± 2.6458  
 

Two solutions were found :