0=x^4-8x^2+8

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


Simplifying
0 = x4 + -8x2 + 8

Reorder the terms:
0 = 8 + -8x2 + x4

Solving
0 = 8 + -8x2 + x4

Solving for variable 'x'.

Combine like terms: 0 + -8 = -8
-8 + 8x2 + -1x4 = 8 + -8x2 + x4 + -8 + 8x2 + -1x4

Reorder the terms:
-8 + 8x2 + -1x4 = 8 + -8 + -8x2 + 8x2 + x4 + -1x4

Combine like terms: 8 + -8 = 0
-8 + 8x2 + -1x4 = 0 + -8x2 + 8x2 + x4 + -1x4
-8 + 8x2 + -1x4 = -8x2 + 8x2 + x4 + -1x4

Combine like terms: -8x2 + 8x2 = 0
-8 + 8x2 + -1x4 = 0 + x4 + -1x4
-8 + 8x2 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-8 + 8x2 + -1x4 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
8 + -8x2 + x4 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -8x2 + -8 + x4 = 0 + -8

Reorder the terms:
8 + -8 + -8x2 + x4 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -8x2 + x4 = 0 + -8
-8x2 + x4 = 0 + -8

Combine like terms: 0 + -8 = -8
-8x2 + x4 = -8

The x term is -8x2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x2 + 16 + x4 = -8 + 16

Reorder the terms:
16 + -8x2 + x4 = -8 + 16

Combine like terms: -8 + 16 = 8
16 + -8x2 + x4 = 8

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x2 + -4) equal to 2.828427125 and -2.828427125.

Subproblem 1

x2 + -4 = 2.828427125 Simplifying x2 + -4 = 2.828427125 Reorder the terms: -4 + x2 = 2.828427125 Solving -4 + x2 = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 2.828427125 + 4 x2 = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 x2 = 6.828427125 Simplifying x2 = 6.828427125 Take the square root of each side: x = {-2.61312593, 2.61312593}

Subproblem 2

x2 + -4 = -2.828427125 Simplifying x2 + -4 = -2.828427125 Reorder the terms: -4 + x2 = -2.828427125 Solving -4 + x2 = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x2 = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -2.828427125 + 4 x2 = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 x2 = 1.171572875 Simplifying x2 = 1.171572875 Take the square root of each side: x = {-1.0823922, 1.0823922}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.61312593, 2.61312593, -1.0823922, 1.0823922}

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