x^4-8x^2+14=0

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Solution for x^4-8x^2+14=0 equation:


Simplifying
x4 + -8x2 + 14 = 0

Reorder the terms:
14 + -8x2 + x4 = 0

Solving
14 + -8x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-14' to each side of the equation.
14 + -8x2 + -14 + x4 = 0 + -14

Reorder the terms:
14 + -14 + -8x2 + x4 = 0 + -14

Combine like terms: 14 + -14 = 0
0 + -8x2 + x4 = 0 + -14
-8x2 + x4 = 0 + -14

Combine like terms: 0 + -14 = -14
-8x2 + x4 = -14

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 = -14 + 16

Reorder the terms:
16 + -8x2 + x4 = -14 + 16

Combine like terms: -14 + 16 = 2
16 + -8x2 + x4 = 2

Factor a perfect square on the left side:
(x2 + -4)(x2 + -4) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x2 + -4) equal to 1.414213562 and -1.414213562.

Subproblem 1

x2 + -4 = 1.414213562 Simplifying x2 + -4 = 1.414213562 Reorder the terms: -4 + x2 = 1.414213562 Solving -4 + x2 = 1.414213562 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 = 1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = 1.414213562 + 4 x2 = 1.414213562 + 4 Combine like terms: 1.414213562 + 4 = 5.414213562 x2 = 5.414213562 Simplifying x2 = 5.414213562 Take the square root of each side: x = {-2.32684627, 2.32684627}

Subproblem 2

x2 + -4 = -1.414213562 Simplifying x2 + -4 = -1.414213562 Reorder the terms: -4 + x2 = -1.414213562 Solving -4 + x2 = -1.414213562 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 = -1.414213562 + 4 Combine like terms: -4 + 4 = 0 0 + x2 = -1.414213562 + 4 x2 = -1.414213562 + 4 Combine like terms: -1.414213562 + 4 = 2.585786438 x2 = 2.585786438 Simplifying x2 = 2.585786438 Take the square root of each side: x = {-1.608038071, 1.608038071}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.32684627, 2.32684627, -1.608038071, 1.608038071}

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