0=7(x^4)-6(x^2)+1

Solution for 0=7(x^4)-6(x^2)+1 equation:

Simplifying
0 = 7(x4) + -6(x2) + 1

Reorder the terms:
0 = 1 + -6x2 + 7x4

Solving
0 = 1 + -6x2 + 7x4

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 6x2 + -7x4 = 1 + -6x2 + 7x4 + -1 + 6x2 + -7x4

Reorder the terms:
-1 + 6x2 + -7x4 = 1 + -1 + -6x2 + 6x2 + 7x4 + -7x4

Combine like terms: 1 + -1 = 0
-1 + 6x2 + -7x4 = 0 + -6x2 + 6x2 + 7x4 + -7x4
-1 + 6x2 + -7x4 = -6x2 + 6x2 + 7x4 + -7x4

Combine like terms: -6x2 + 6x2 = 0
-1 + 6x2 + -7x4 = 0 + 7x4 + -7x4
-1 + 6x2 + -7x4 = 7x4 + -7x4

Combine like terms: 7x4 + -7x4 = 0
-1 + 6x2 + -7x4 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
0.1428571429 + -0.8571428571x2 + x4 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -0.8571428571x2 + -0.1428571429 + x4 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + -0.8571428571x2 + x4 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + -0.8571428571x2 + x4 = 0 + -0.1428571429
-0.8571428571x2 + x4 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
-0.8571428571x2 + x4 = -0.1428571429

The x term is -0.8571428571x2.  Take half its coefficient (-0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
-0.8571428571x2 + 0.1836734694 + x4 = -0.1428571429 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571x2 + x4 = -0.1428571429 + 0.1836734694

Combine like terms: -0.1428571429 + 0.1836734694 = 0.0408163265
0.1836734694 + -0.8571428571x2 + x4 = 0.0408163265

Factor a perfect square on the left side:
(x2 + -0.4285714286)(x2 + -0.4285714286) = 0.0408163265

Calculate the square root of the right side: 0.202030509

Break this problem into two subproblems by setting 
(x2 + -0.4285714286) equal to 0.202030509 and -0.202030509.

Subproblem 1
x2 + -0.4285714286 = 0.202030509

Simplifying
x2 + -0.4285714286 = 0.202030509

Reorder the terms:
-0.4285714286 + x2 = 0.202030509

Solving
-0.4285714286 + x2 = 0.202030509

Solving for variable 'x'.

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

Add '0.4285714286' to each side of the equation.
-0.4285714286 + 0.4285714286 + x2 = 0.202030509 + 0.4285714286

Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000
0.0000000000 + x2 = 0.202030509 + 0.4285714286
x2 = 0.202030509 + 0.4285714286

Combine like terms: 0.202030509 + 0.4285714286 = 0.6306019376
x2 = 0.6306019376

Simplifying
x2 = 0.6306019376

Take the square root of each side:
x = {-0.794104488, 0.794104488}

Subproblem 2
x2 + -0.4285714286 = -0.202030509

Simplifying
x2 + -0.4285714286 = -0.202030509

Reorder the terms:
-0.4285714286 + x2 = -0.202030509

Solving
-0.4285714286 + x2 = -0.202030509

Solving for variable 'x'.

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

Add '0.4285714286' to each side of the equation.
-0.4285714286 + 0.4285714286 + x2 = -0.202030509 + 0.4285714286

Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000
0.0000000000 + x2 = -0.202030509 + 0.4285714286
x2 = -0.202030509 + 0.4285714286

Combine like terms: -0.202030509 + 0.4285714286 = 0.2265409196
x2 = 0.2265409196

Simplifying
x2 = 0.2265409196

Take the square root of each side:
x = {-0.475963149, 0.475963149}

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.794104488, 0.794104488, -0.475963149, 0.475963149}