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

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


Simplifying
x4 + -5x2 + 2 = 0

Reorder the terms:
2 + -5x2 + x4 = 0

Solving
2 + -5x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -5x2 + -2 + x4 = 0 + -2

Reorder the terms:
2 + -2 + -5x2 + x4 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -5x2 + x4 = 0 + -2
-5x2 + x4 = 0 + -2

Combine like terms: 0 + -2 = -2
-5x2 + x4 = -2

The x term is -5x2.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x2 + 6.25 + x4 = -2 + 6.25

Reorder the terms:
6.25 + -5x2 + x4 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + -5x2 + x4 = 4.25

Factor a perfect square on the left side:
(x2 + -2.5)(x2 + -2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x2 + -2.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x2 + -2.5 = 2.061552813 Simplifying x2 + -2.5 = 2.061552813 Reorder the terms: -2.5 + x2 = 2.061552813 Solving -2.5 + x2 = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = 2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = 2.061552813 + 2.5 x2 = 2.061552813 + 2.5 Combine like terms: 2.061552813 + 2.5 = 4.561552813 x2 = 4.561552813 Simplifying x2 = 4.561552813 Take the square root of each side: x = {-2.135779205, 2.135779205}

Subproblem 2

x2 + -2.5 = -2.061552813 Simplifying x2 + -2.5 = -2.061552813 Reorder the terms: -2.5 + x2 = -2.061552813 Solving -2.5 + x2 = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x2 = -2.061552813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x2 = -2.061552813 + 2.5 x2 = -2.061552813 + 2.5 Combine like terms: -2.061552813 + 2.5 = 0.438447187 x2 = 0.438447187 Simplifying x2 = 0.438447187 Take the square root of each side: x = {-0.662153447, 0.662153447}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.135779205, 2.135779205, -0.662153447, 0.662153447}

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