Solution - Equations reducible to quadratic form

Rearrange:

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

                     x^4-(12*x^2-8)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (x4) -  ((22•3x2) -  8)  = 0 
 Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  x⁴-12x²+8 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  -12x²  its coefficient is  -12 .
The last term, "the constant", is  +8 

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

Step-2 : Find two factors of  8  whose sum equals the coefficient of the middle term, which is   -12 .

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

Equation at the end of step  2  :

  x4 - 12x2 + 8  = 0 

Step  3  :

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 3.1     Solve   x⁴-12x²+8 = 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 :
 w²-12w+8 = 0

Solving this new equation using the quadratic formula we get two real solutions :
   11.2915  or   0.7085

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
   x⁴-12x²+8 = 0
  are either : 
   x =√11.292 = 3.36028  or :
   x =√11.292 = -3.36028  or :
   x =√ 0.708 = 0.84172  or :
   x =√ 0.708 = -0.84172

Four solutions were found :

1.  x =√ 0.708 = -0.84172
2.  x =√ 0.708 = 0.84172
3.  x =√11.292 = -3.36028
4.  x =√11.292 = 3.36028