x^4+6=6x^2

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Solution for x^4+6=6x^2 equation:


Simplifying
x4 + 6 = 6x2

Reorder the terms:
6 + x4 = 6x2

Solving
6 + x4 = 6x2

Solving for variable 'x'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = -6 + 9

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

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

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

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x2 + -3) equal to 1.732050808 and -1.732050808.

Subproblem 1

x2 + -3 = 1.732050808 Simplifying x2 + -3 = 1.732050808 Reorder the terms: -3 + x2 = 1.732050808 Solving -3 + x2 = 1.732050808 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 = 1.732050808 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = 1.732050808 + 3 x2 = 1.732050808 + 3 Combine like terms: 1.732050808 + 3 = 4.732050808 x2 = 4.732050808 Simplifying x2 = 4.732050808 Take the square root of each side: x = {-2.175327747, 2.175327747}

Subproblem 2

x2 + -3 = -1.732050808 Simplifying x2 + -3 = -1.732050808 Reorder the terms: -3 + x2 = -1.732050808 Solving -3 + x2 = -1.732050808 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 = -1.732050808 + 3 Combine like terms: -3 + 3 = 0 0 + x2 = -1.732050808 + 3 x2 = -1.732050808 + 3 Combine like terms: -1.732050808 + 3 = 1.267949192 x2 = 1.267949192 Simplifying x2 = 1.267949192 Take the square root of each side: x = {-1.1260325, 1.1260325}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.175327747, 2.175327747, -1.1260325, 1.1260325}

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