16x^4-41x^2=25

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Solution for 16x^4-41x^2=25 equation:


Simplifying
16x4 + -41x2 = 25

Reorder the terms:
-41x2 + 16x4 = 25

Solving
-41x2 + 16x4 = 25

Solving for variable 'x'.

Reorder the terms:
-25 + -41x2 + 16x4 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + -41x2 + 16x4 = 0

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

Divide each side by '16'.
-1.5625 + -2.5625x2 + x4 = 0

Move the constant term to the right:

Add '1.5625' to each side of the equation.
-1.5625 + -2.5625x2 + 1.5625 + x4 = 0 + 1.5625

Reorder the terms:
-1.5625 + 1.5625 + -2.5625x2 + x4 = 0 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + -2.5625x2 + x4 = 0 + 1.5625
-2.5625x2 + x4 = 0 + 1.5625

Combine like terms: 0 + 1.5625 = 1.5625
-2.5625x2 + x4 = 1.5625

The x term is -2.5625x2.  Take half its coefficient (-1.28125).
Square it (1.641601563) and add it to both sides.

Add '1.641601563' to each side of the equation.
-2.5625x2 + 1.641601563 + x4 = 1.5625 + 1.641601563

Reorder the terms:
1.641601563 + -2.5625x2 + x4 = 1.5625 + 1.641601563

Combine like terms: 1.5625 + 1.641601563 = 3.204101563
1.641601563 + -2.5625x2 + x4 = 3.204101563

Factor a perfect square on the left side:
(x2 + -1.28125)(x2 + -1.28125) = 3.204101563

Calculate the square root of the right side: 1.790000437

Break this problem into two subproblems by setting 
(x2 + -1.28125) equal to 1.790000437 and -1.790000437.

Subproblem 1

x2 + -1.28125 = 1.790000437 Simplifying x2 + -1.28125 = 1.790000437 Reorder the terms: -1.28125 + x2 = 1.790000437 Solving -1.28125 + x2 = 1.790000437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + x2 = 1.790000437 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + x2 = 1.790000437 + 1.28125 x2 = 1.790000437 + 1.28125 Combine like terms: 1.790000437 + 1.28125 = 3.071250437 x2 = 3.071250437 Simplifying x2 = 3.071250437 Take the square root of each side: x = {-1.752498342, 1.752498342}

Subproblem 2

x2 + -1.28125 = -1.790000437 Simplifying x2 + -1.28125 = -1.790000437 Reorder the terms: -1.28125 + x2 = -1.790000437 Solving -1.28125 + x2 = -1.790000437 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + x2 = -1.790000437 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + x2 = -1.790000437 + 1.28125 x2 = -1.790000437 + 1.28125 Combine like terms: -1.790000437 + 1.28125 = -0.508750437 x2 = -0.508750437 Simplifying x2 = -0.508750437 Reorder the terms: 0.508750437 + x2 = -0.508750437 + 0.508750437 Combine like terms: -0.508750437 + 0.508750437 = 0.000000000 0.508750437 + 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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