4z^4-8z^2+3=0

Solution for 4z^4-8z^2+3=0 equation:

Simplifying
4z4 + -8z2 + 3 = 0

Reorder the terms:
3 + -8z2 + 4z4 = 0

Solving
3 + -8z2 + 4z4 = 0

Solving for variable 'z'.

Factor a trinomial.
(1 + -2z2)(3 + -2z2) = 0

Subproblem 1
Set the factor '(1 + -2z2)' equal to zero and attempt to solve:

Simplifying
1 + -2z2 = 0

Solving
1 + -2z2 = 0

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

Add '-1' to each side of the equation.
1 + -1 + -2z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2z2 = 0 + -1
-2z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2z2 = -1

Divide each side by '-2'.
z2 = 0.5

Simplifying
z2 = 0.5

Take the square root of each side:
z = {-0.707106781, 0.707106781}
Subproblem 2
Set the factor '(3 + -2z2)' equal to zero and attempt to solve:

Simplifying
3 + -2z2 = 0

Solving
3 + -2z2 = 0

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

Add '-3' to each side of the equation.
3 + -3 + -2z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -2z2 = 0 + -3
-2z2 = 0 + -3

Combine like terms: 0 + -3 = -3
-2z2 = -3

Divide each side by '-2'.
z2 = 1.5

Simplifying
z2 = 1.5

Take the square root of each side:
z = {-1.224744871, 1.224744871}
Solution
z = {-0.707106781, 0.707106781, -1.224744871, 1.224744871}