Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (0 -  (2x26 • x)) -  7  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -2x²⁷ - 7  =   -1 • (2x²⁷ + 7) 

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  2x²⁷ + 7 

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 :  2  is not a cube !!

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

Equation at the end of step  3  :

  -2x27 - 7  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    -2x²⁷-7 = 0 

 Add  7  to both sides of the equation : 
                      -2x²⁷ = 7
Multiply both sides of the equation by (-1) :  2x²⁷ = -7


Divide both sides of the equation by 2:
                     x²⁷ = -7/2 = -3.500
                     x  =  27th root of (-7/2) 

 Negative numbers have real 27th roots.
 27th root of (-7/2) = ²⁷√ -1• 7/2  = ²⁷√ -1 • ²⁷√ 7/2  =(-1)•²⁷√ 7/2 

The equation has one real solution, a negative number This solution is  x = 27th root of (-3.500) = -1.04749

One solution was found :