Solving nonlinear equations for both real and imaginary numbers

Question

9x^4-28x^2+3=0

Answer 1

9x⁴ - 28x² + 3 = 0

 

This is a "quadratic-like" equation.  Let w = x².  When you substitute that in, the equation becomes:

 

     9w² - 28w + 3 = 0

 

which factors to:

 

     9w² - 27w - w + 3

     9w(w-3) - (w-3)

     (9w - 1)(w - 3)

 

     w = 1/9 and 3

 

Since w = x²:

 

     x = √w = ±1/3 and ±√3

Answer 2

9x⁴ - 28x² + 3 = 0

 

Since the c value is positive, the sign of both factors must be the same. Since there is a minus sign in front of the b value, both factors must have a minus sign.

 

(9x² - 1)(x² - 3) = 0

 

9x² - 1 = 0

 

(3x+1)(3x-1) = 0

 

3x = -1 x = -1/3

3x = 1  x = 1/3

 

x² - 3 =0

 

x² = 3

 

x = +/- √3