5x^4-10x^2+1=0

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Solution for 5x^4-10x^2+1=0 equation:


Simplifying
5x4 + -10x2 + 1 = 0

Reorder the terms:
1 + -10x2 + 5x4 = 0

Solving
1 + -10x2 + 5x4 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.2 + -2x2 + x4 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -2x2 + -0.2 + x4 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -2x2 + x4 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -2x2 + x4 = 0 + -0.2
-2x2 + x4 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-2x2 + x4 = -0.2

The x term is -2x2.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x2 + 1 + x4 = -0.2 + 1

Reorder the terms:
1 + -2x2 + x4 = -0.2 + 1

Combine like terms: -0.2 + 1 = 0.8
1 + -2x2 + x4 = 0.8

Factor a perfect square on the left side:
(x2 + -1)(x2 + -1) = 0.8

Calculate the square root of the right side: 0.894427191

Break this problem into two subproblems by setting 
(x2 + -1) equal to 0.894427191 and -0.894427191.

Subproblem 1

x2 + -1 = 0.894427191 Simplifying x2 + -1 = 0.894427191 Reorder the terms: -1 + x2 = 0.894427191 Solving -1 + x2 = 0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = 0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = 0.894427191 + 1 x2 = 0.894427191 + 1 Combine like terms: 0.894427191 + 1 = 1.894427191 x2 = 1.894427191 Simplifying x2 = 1.894427191 Take the square root of each side: x = {-1.37638192, 1.37638192}

Subproblem 2

x2 + -1 = -0.894427191 Simplifying x2 + -1 = -0.894427191 Reorder the terms: -1 + x2 = -0.894427191 Solving -1 + x2 = -0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x2 = -0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x2 = -0.894427191 + 1 x2 = -0.894427191 + 1 Combine like terms: -0.894427191 + 1 = 0.105572809 x2 = 0.105572809 Simplifying x2 = 0.105572809 Take the square root of each side: x = {-0.324919696, 0.324919696}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.37638192, 1.37638192, -0.324919696, 0.324919696}

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