X^4-2X^2=1

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Solution for X^4-2X^2=1 equation:


Simplifying
X4 + -2X2 = 1

Reorder the terms:
-2X2 + X4 = 1

Solving
-2X2 + X4 = 1

Solving for variable 'X'.

Reorder the terms:
-1 + -2X2 + X4 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -2X2 + X4 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2X2 + 1 + X4 = 0 + 1

Reorder the terms:
-1 + 1 + -2X2 + X4 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2X2 + X4 = 0 + 1
-2X2 + X4 = 0 + 1

Combine like terms: 0 + 1 = 1
-2X2 + X4 = 1

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 = 1 + 1

Reorder the terms:
1 + -2X2 + X4 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2X2 + X4 = 2

Factor a perfect square on the left side:
(X2 + -1)(X2 + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(X2 + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1

X2 + -1 = 1.414213562 Simplifying X2 + -1 = 1.414213562 Reorder the terms: -1 + X2 = 1.414213562 Solving -1 + X2 = 1.414213562 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 = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + X2 = 1.414213562 + 1 X2 = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 X2 = 2.414213562 Simplifying X2 = 2.414213562 Take the square root of each side: X = {-1.553773974, 1.553773974}

Subproblem 2

X2 + -1 = -1.414213562 Simplifying X2 + -1 = -1.414213562 Reorder the terms: -1 + X2 = -1.414213562 Solving -1 + X2 = -1.414213562 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 = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + X2 = -1.414213562 + 1 X2 = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 X2 = -0.414213562 Simplifying X2 = -0.414213562 Reorder the terms: 0.414213562 + X2 = -0.414213562 + 0.414213562 Combine like terms: -0.414213562 + 0.414213562 = 0.000000000 0.414213562 + 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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