Solution - Equations reducible to quadratic form

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((x6) +  2x3) -  27  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  x⁶+2x³-27 

The first term is,  x⁶  its coefficient is  1 .
The middle term is,  +2x³  its coefficient is  2 .
The last term, "the constant", is  -27 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -27 = -27 

Step-2 : Find two factors of  -27  whose sum equals the coefficient of the middle term, which is   2 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  2  :

  x6 + 2x3 - 27  = 0 

Step  3  :

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 3.1     Solve   x⁶+2x³-27 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x³  transforms the equation into :
 w²+2w-27 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   4.2915  or  -6.2915

Now that we know the value(s) of  w , we can calculate  x  since  x  is  ∛ w  

Doing just this we discover that the solutions of
   x⁶+2x³-27 = 0
  are either : 
     x = ∛ 4.292 = 1.6251
   or:
   x = ∛-6.292 = -1.8461

Two solutions were found :

1.  x = ∛-6.292 = -1.8461
2.    x = ∛ 4.292 = 1.6251