X^4-5X^2=-1

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Solution for X^4-5X^2=-1 equation:


Simplifying
X4 + -5X2 = -1

Reorder the terms:
-5X2 + X4 = -1

Solving
-5X2 + X4 = -1

Solving for variable 'X'.

Reorder the terms:
1 + -5X2 + X4 = -1 + 1

Combine like terms: -1 + 1 = 0
1 + -5X2 + X4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5X2 + -1 + X4 = 0 + -1

Reorder the terms:
1 + -1 + -5X2 + X4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5X2 + X4 = 0 + -1
-5X2 + X4 = 0 + -1

Combine like terms: 0 + -1 = -1
-5X2 + X4 = -1

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 = -1 + 6.25

Reorder the terms:
6.25 + -5X2 + X4 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5X2 + X4 = 5.25

Factor a perfect square on the left side:
(X2 + -2.5)(X2 + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(X2 + -2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

X2 + -2.5 = 2.291287847 Simplifying X2 + -2.5 = 2.291287847 Reorder the terms: -2.5 + X2 = 2.291287847 Solving -2.5 + X2 = 2.291287847 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 = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + X2 = 2.291287847 + 2.5 X2 = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 X2 = 4.791287847 Simplifying X2 = 4.791287847 Take the square root of each side: X = {-2.188901059, 2.188901059}

Subproblem 2

X2 + -2.5 = -2.291287847 Simplifying X2 + -2.5 = -2.291287847 Reorder the terms: -2.5 + X2 = -2.291287847 Solving -2.5 + X2 = -2.291287847 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 = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + X2 = -2.291287847 + 2.5 X2 = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 X2 = 0.208712153 Simplifying X2 = 0.208712153 Take the square root of each side: X = {-0.456850252, 0.456850252}

Solution

The solution to the problem is based on the solutions from the subproblems. X = {-2.188901059, 2.188901059, -0.456850252, 0.456850252}

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