x^4-6x^2+3=0

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Solution for x^4-6x^2+3=0 equation:


Simplifying
x4 + -6x2 + 3 = 0

Reorder the terms:
3 + -6x2 + x4 = 0

Solving
3 + -6x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -6x2 + -3 + x4 = 0 + -3

Reorder the terms:
3 + -3 + -6x2 + x4 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -6x2 + x4 = 0 + -3
-6x2 + x4 = 0 + -3

Combine like terms: 0 + -3 = -3
-6x2 + x4 = -3

The x term is -6x2.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x2 + 9 + x4 = -3 + 9

Reorder the terms:
9 + -6x2 + x4 = -3 + 9

Combine like terms: -3 + 9 = 6
9 + -6x2 + x4 = 6

Factor a perfect square on the left side:
(x2 + -3)(x2 + -3) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x2 + -3) equal to 2.449489743 and -2.449489743.

Subproblem 1

x2 + -3 = 2.449489743 Simplifying x2 + -3 = 2.449489743 Reorder the terms: -3 + x2 = 2.449489743 Solving -3 + x2 = 2.449489743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = 2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = 2.449489743 + 3 x2 = 2.449489743 + 3 Combine like terms: 2.449489743 + 3 = 5.449489743 x2 = 5.449489743 Simplifying x2 = 5.449489743 Take the square root of each side: x = {-2.334414218, 2.334414218}

Subproblem 2

x2 + -3 = -2.449489743 Simplifying x2 + -3 = -2.449489743 Reorder the terms: -3 + x2 = -2.449489743 Solving -3 + x2 = -2.449489743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x2 = -2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = -2.449489743 + 3 x2 = -2.449489743 + 3 Combine like terms: -2.449489743 + 3 = 0.550510257 x2 = 0.550510257 Simplifying x2 = 0.550510257 Take the square root of each side: x = {-0.741963784, 0.741963784}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.334414218, 2.334414218, -0.741963784, 0.741963784}

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