Solution - Equations reducible to quadratic form

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((x5) -  22x3) -  2x  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x⁵ - 4x³ - 2x  =   x • (x⁴ - 4x² - 2) 

Trying to factor by splitting the middle term

 3.2     Factoring  x⁴ - 4x² - 2 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  -4x²  its coefficient is  -4 .
The last term, "the constant", is  -2 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -2 = -2 

Step-2 : Find two factors of  -2  whose sum equals the coefficient of the middle term, which is   -4 .

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

Equation at the end of step  3  :

  x • (x4 - 4x2 - 2)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 4.2      Solve  :    x = 0 

  Solution is  x = 0

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 4.3     Solve   x⁴-4x²-2 = 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 = x²  transforms the equation into :
 w²-4w-2 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   4.4495  or  -0.4495

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
   x⁴-4x²-2 = 0
  are either : 
   x =√ 4.449 = 2.10938  or :
   x =√ 4.449 = -2.10938  or :
   x =√-0.449 = 0.0 + 0.67044 i  or :
   x =√-0.449 = 0.0 - 0.67044 i

5 solutions were found :

1.  x =√-0.449 = 0.0 - 0.67044 i
2.  x =√-0.449 = 0.0 + 0.67044 i
3.  x =√ 4.449 = -2.10938
4.  x =√ 4.449 = 2.10938
5.  x = 0