x^4-10x^2+8=0

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Solution for x^4-10x^2+8=0 equation:


Simplifying
x4 + -10x2 + 8 = 0

Reorder the terms:
8 + -10x2 + x4 = 0

Solving
8 + -10x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -10x2 + -8 + x4 = 0 + -8

Reorder the terms:
8 + -8 + -10x2 + x4 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -10x2 + x4 = 0 + -8
-10x2 + x4 = 0 + -8

Combine like terms: 0 + -8 = -8
-10x2 + x4 = -8

The x term is -10x2.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x2 + 25 + x4 = -8 + 25

Reorder the terms:
25 + -10x2 + x4 = -8 + 25

Combine like terms: -8 + 25 = 17
25 + -10x2 + x4 = 17

Factor a perfect square on the left side:
(x2 + -5)(x2 + -5) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(x2 + -5) equal to 4.123105626 and -4.123105626.

Subproblem 1

x2 + -5 = 4.123105626 Simplifying x2 + -5 = 4.123105626 Reorder the terms: -5 + x2 = 4.123105626 Solving -5 + x2 = 4.123105626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x2 = 4.123105626 + 5 Combine like terms: -5 + 5 = 0 0 + x2 = 4.123105626 + 5 x2 = 4.123105626 + 5 Combine like terms: 4.123105626 + 5 = 9.123105626 x2 = 9.123105626 Simplifying x2 = 9.123105626 Take the square root of each side: x = {-3.020447918, 3.020447918}

Subproblem 2

x2 + -5 = -4.123105626 Simplifying x2 + -5 = -4.123105626 Reorder the terms: -5 + x2 = -4.123105626 Solving -5 + x2 = -4.123105626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x2 = -4.123105626 + 5 Combine like terms: -5 + 5 = 0 0 + x2 = -4.123105626 + 5 x2 = -4.123105626 + 5 Combine like terms: -4.123105626 + 5 = 0.876894374 x2 = 0.876894374 Simplifying x2 = 0.876894374 Take the square root of each side: x = {-0.936426385, 0.936426385}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.020447918, 3.020447918, -0.936426385, 0.936426385}

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