x^4+x^2=684

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Solution for x^4+x^2=684 equation:


Simplifying
x4 + x2 = 684

Reorder the terms:
x2 + x4 = 684

Solving
x2 + x4 = 684

Solving for variable 'x'.

Reorder the terms:
-684 + x2 + x4 = 684 + -684

Combine like terms: 684 + -684 = 0
-684 + x2 + x4 = 0

Begin completing the square.

Move the constant term to the right:

Add '684' to each side of the equation.
-684 + x2 + 684 + x4 = 0 + 684

Reorder the terms:
-684 + 684 + x2 + x4 = 0 + 684

Combine like terms: -684 + 684 = 0
0 + x2 + x4 = 0 + 684
x2 + x4 = 0 + 684

Combine like terms: 0 + 684 = 684
x2 + x4 = 684

The x term is x2.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x2 + 0.25 + x4 = 684 + 0.25

Reorder the terms:
0.25 + x2 + x4 = 684 + 0.25

Combine like terms: 684 + 0.25 = 684.25
0.25 + x2 + x4 = 684.25

Factor a perfect square on the left side:
(x2 + 0.5)(x2 + 0.5) = 684.25

Calculate the square root of the right side: 26.158172719

Break this problem into two subproblems by setting 
(x2 + 0.5) equal to 26.158172719 and -26.158172719.

Subproblem 1

x2 + 0.5 = 26.158172719 Simplifying x2 + 0.5 = 26.158172719 Reorder the terms: 0.5 + x2 = 26.158172719 Solving 0.5 + x2 = 26.158172719 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x2 = 26.158172719 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x2 = 26.158172719 + -0.5 x2 = 26.158172719 + -0.5 Combine like terms: 26.158172719 + -0.5 = 25.658172719 x2 = 25.658172719 Simplifying x2 = 25.658172719 Take the square root of each side: x = {-5.065389691, 5.065389691}

Subproblem 2

x2 + 0.5 = -26.158172719 Simplifying x2 + 0.5 = -26.158172719 Reorder the terms: 0.5 + x2 = -26.158172719 Solving 0.5 + x2 = -26.158172719 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x2 = -26.158172719 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x2 = -26.158172719 + -0.5 x2 = -26.158172719 + -0.5 Combine like terms: -26.158172719 + -0.5 = -26.658172719 x2 = -26.658172719 Simplifying x2 = -26.658172719 Reorder the terms: 26.658172719 + x2 = -26.658172719 + 26.658172719 Combine like terms: -26.658172719 + 26.658172719 = 0.000000000 26.658172719 + x2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.

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