4x^4-16.25x^2+4=0

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Solution for 4x^4-16.25x^2+4=0 equation:


Simplifying
4x4 + -16.25x2 + 4 = 0

Reorder the terms:
4 + -16.25x2 + 4x4 = 0

Solving
4 + -16.25x2 + 4x4 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
1 + -4.0625x2 + x4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4.0625x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -4.0625x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4.0625x2 + x4 = 0 + -1
-4.0625x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-4.0625x2 + x4 = -1

The x term is -4.0625x2.  Take half its coefficient (-2.03125).
Square it (4.125976563) and add it to both sides.

Add '4.125976563' to each side of the equation.
-4.0625x2 + 4.125976563 + x4 = -1 + 4.125976563

Reorder the terms:
4.125976563 + -4.0625x2 + x4 = -1 + 4.125976563

Combine like terms: -1 + 4.125976563 = 3.125976563
4.125976563 + -4.0625x2 + x4 = 3.125976563

Factor a perfect square on the left side:
(x2 + -2.03125)(x2 + -2.03125) = 3.125976563

Calculate the square root of the right side: 1.768043145

Break this problem into two subproblems by setting 
(x2 + -2.03125) equal to 1.768043145 and -1.768043145.

Subproblem 1

x2 + -2.03125 = 1.768043145 Simplifying x2 + -2.03125 = 1.768043145 Reorder the terms: -2.03125 + x2 = 1.768043145 Solving -2.03125 + x2 = 1.768043145 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + x2 = 1.768043145 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + x2 = 1.768043145 + 2.03125 x2 = 1.768043145 + 2.03125 Combine like terms: 1.768043145 + 2.03125 = 3.799293145 x2 = 3.799293145 Simplifying x2 = 3.799293145 Take the square root of each side: x = {-1.949177556, 1.949177556}

Subproblem 2

x2 + -2.03125 = -1.768043145 Simplifying x2 + -2.03125 = -1.768043145 Reorder the terms: -2.03125 + x2 = -1.768043145 Solving -2.03125 + x2 = -1.768043145 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + x2 = -1.768043145 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + x2 = -1.768043145 + 2.03125 x2 = -1.768043145 + 2.03125 Combine like terms: -1.768043145 + 2.03125 = 0.263206855 x2 = 0.263206855 Simplifying x2 = 0.263206855 Take the square root of each side: x = {-0.513036894, 0.513036894}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.949177556, 1.949177556, -0.513036894, 0.513036894}

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