Solution - Nonlinear equations

Rearrange:

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

                     a^2+16^2-(25^2)=0 

Step by step solution :

Step  1  :

 1.1     25 = 52

(25)² = (5²)² = 5⁴

Equation at the end of step  1  :

  ((a2) +  (162)) -  54  = 0 

Step  2  :

 2.1     16 = 24

(16)² = (2⁴)² = 2⁸

Equation at the end of step  2  :

  ((a2) +  28) -  54  = 0 

Step  3  :

Trying to factor as a Difference of Squares :

 3.1      Factoring:  a²-369 

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 : 369 is not a square !!

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

Equation at the end of step  3  :

  a2 - 369  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    a²-369 = 0 

 Add  369  to both sides of the equation : 
                      a² = 369
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      a  =  ± √ 369  

 Can  √ 369 be simplified ?

Yes!   The prime factorization of  369   is
   3•3•41 
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).

√ 369   =  √ 3•3•41   =
                ±  3 • √ 41

The equation has two real solutions  
 These solutions are  a = 3 • ± √41 = ± 19.2094  
 

Two solutions were found :