5x^4-12x^2+3=0

Solution for 5x^4-12x^2+3=0 equation:

Simplifying
5x4 + -12x2 + 3 = 0

Reorder the terms:
3 + -12x2 + 5x4 = 0

Solving
3 + -12x2 + 5x4 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
0.6 + -2.4x2 + x4 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + -2.4x2 + -0.6 + x4 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + -2.4x2 + x4 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + -2.4x2 + x4 = 0 + -0.6
-2.4x2 + x4 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
-2.4x2 + x4 = -0.6

The x term is -2.4x2.  Take half its coefficient (-1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
-2.4x2 + 1.44 + x4 = -0.6 + 1.44

Reorder the terms:
1.44 + -2.4x2 + x4 = -0.6 + 1.44

Combine like terms: -0.6 + 1.44 = 0.84
1.44 + -2.4x2 + x4 = 0.84

Factor a perfect square on the left side:
(x2 + -1.2)(x2 + -1.2) = 0.84

Calculate the square root of the right side: 0.916515139

Break this problem into two subproblems by setting 
(x2 + -1.2) equal to 0.916515139 and -0.916515139.

Subproblem 1
x2 + -1.2 = 0.916515139

Simplifying
x2 + -1.2 = 0.916515139

Reorder the terms:
-1.2 + x2 = 0.916515139

Solving
-1.2 + x2 = 0.916515139

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.2' to each side of the equation.
-1.2 + 1.2 + x2 = 0.916515139 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + x2 = 0.916515139 + 1.2
x2 = 0.916515139 + 1.2

Combine like terms: 0.916515139 + 1.2 = 2.116515139
x2 = 2.116515139

Simplifying
x2 = 2.116515139

Take the square root of each side:
x = {-1.454824779, 1.454824779}

Subproblem 2
x2 + -1.2 = -0.916515139

Simplifying
x2 + -1.2 = -0.916515139

Reorder the terms:
-1.2 + x2 = -0.916515139

Solving
-1.2 + x2 = -0.916515139

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.2' to each side of the equation.
-1.2 + 1.2 + x2 = -0.916515139 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + x2 = -0.916515139 + 1.2
x2 = -0.916515139 + 1.2

Combine like terms: -0.916515139 + 1.2 = 0.283484861
x2 = 0.283484861

Simplifying
x2 = 0.283484861

Take the square root of each side:
x = {-0.532432964, 0.532432964}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.454824779, 1.454824779, -0.532432964, 0.532432964}