Solution - Equations reducible to quadratic form

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x4"   was replaced by   "x^4". 

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     4*x^2-8*x^4-(1)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((4 • (x2)) -  23x4) -  1  = 0 
 Step  2  :

Equation at the end of step  2  :

  (22x2 -  23x4) -  1  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   -8x⁴ + 4x² - 1  =   -1 • (8x⁴ - 4x² + 1) 

Trying to factor by splitting the middle term

 4.2     Factoring  8x⁴ - 4x² + 1 

The first term is,  8x⁴  its coefficient is  8 .
The middle term is,  -4x²  its coefficient is  -4 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   8 • 1 = 8 

Step-2 : Find two factors of  8  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  4  :

  -8x4 + 4x2 - 1  = 0 

Step  5  :

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 5.1     Solve   -8x⁴+4x²-1 = 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 :
 -8w²+4w-1 = 0

Solving this new equation using the quadratic formula we get two imaginary solutions :
   w = 16.0000 ± -16.0000 i 
Now that we know the value(s) of  w , we can calculate  x  since  x  is  √ w  

Since we are speaking 2nd root, each of the two imaginary solutions of has 2 roots

Tiger finds these roots using de Moivre's Formula

The 2nd roots of  16.000 + -16.000 i   are:

  x = -4.395 + 1.820 i 
  x =  4.395 -1.820 i 

2nd roots of  16.000--16.000 i  :
  x =  4.395  + 1.820 i  
  x = -4.395 - 1.820 i 
 

Four solutions were found :

1.   x = -4.395 - 1.820 i
2.   x = 4.395 + 1.820 i
3.   x = 4.395 -1.820 i
4.   x = -4.395 + 1.820 i