Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (0 -  (32x212 • x)) -  4  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -9x²¹³ - 4  =   -1 • (9x²¹³ + 4) 

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  9x²¹³ + 4 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  9  is not a cube !!

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

Equation at the end of step  3  :

  -9x213 - 4  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    -9x²¹³-4 = 0 

 Add  4  to both sides of the equation : 
                      -9x²¹³ = 4
Multiply both sides of the equation by (-1) :  9x²¹³ = -4


Divide both sides of the equation by 9:
                     x²¹³ = -4/9 = -0.444
                     x  =  213th root of (-4/9) 

 Negative numbers have real 213th roots.
 213th root of (-4/9) = ²¹³√ -1• 4/9  = ²¹³√ -1 • ²¹³√ 4/9  =(-1)•²¹³√ 4/9 

The equation has one real solution, a negative number This solution is  x = 213th root of (-0.444) = -0.99620

One solution was found :