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+13^2-(21^2)=0 

Step by step solution :

Step  1  :

 1.1     21 = 3•7

(21)² = (3•7)² = 3² • 7²

Equation at the end of step  1  :

  ((a2) +  (132)) -  (32•72)  = 0 

Step  2  :

Equation at the end of step  2  :

  ((a2) +  132) -  (32•72)  = 0 

Step  3  :

Trying to factor as a Difference of Squares :

 3.1      Factoring:  a²-272 

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 : 272 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 - 272  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    a²-272 = 0 

 Add  272  to both sides of the equation : 
                      a² = 272
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      a  =  ± √ 272  

 Can  √ 272 be simplified ?

Yes!   The prime factorization of  272   is
   2•2•2•2•17 
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).

√ 272   =  √ 2•2•2•2•17   =2•2•√ 17   =
                ±  4 • √ 17

The equation has two real solutions  
 These solutions are  a = 4 • ± √17 = ± 16.4924  
 

Two solutions were found :