5x^4-401x^2+16=0

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Solution for 5x^4-401x^2+16=0 equation:


Simplifying
5x4 + -401x2 + 16 = 0

Reorder the terms:
16 + -401x2 + 5x4 = 0

Solving
16 + -401x2 + 5x4 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
3.2 + -80.2x2 + x4 = 0

Move the constant term to the right:

Add '-3.2' to each side of the equation.
3.2 + -80.2x2 + -3.2 + x4 = 0 + -3.2

Reorder the terms:
3.2 + -3.2 + -80.2x2 + x4 = 0 + -3.2

Combine like terms: 3.2 + -3.2 = 0.0
0.0 + -80.2x2 + x4 = 0 + -3.2
-80.2x2 + x4 = 0 + -3.2

Combine like terms: 0 + -3.2 = -3.2
-80.2x2 + x4 = -3.2

The x term is -80.2x2.  Take half its coefficient (-40.1).
Square it (1608.01) and add it to both sides.

Add '1608.01' to each side of the equation.
-80.2x2 + 1608.01 + x4 = -3.2 + 1608.01

Reorder the terms:
1608.01 + -80.2x2 + x4 = -3.2 + 1608.01

Combine like terms: -3.2 + 1608.01 = 1604.81
1608.01 + -80.2x2 + x4 = 1604.81

Factor a perfect square on the left side:
(x2 + -40.1)(x2 + -40.1) = 1604.81

Calculate the square root of the right side: 40.06007988

Break this problem into two subproblems by setting 
(x2 + -40.1) equal to 40.06007988 and -40.06007988.

Subproblem 1

x2 + -40.1 = 40.06007988 Simplifying x2 + -40.1 = 40.06007988 Reorder the terms: -40.1 + x2 = 40.06007988 Solving -40.1 + x2 = 40.06007988 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40.1' to each side of the equation. -40.1 + 40.1 + x2 = 40.06007988 + 40.1 Combine like terms: -40.1 + 40.1 = 0.0 0.0 + x2 = 40.06007988 + 40.1 x2 = 40.06007988 + 40.1 Combine like terms: 40.06007988 + 40.1 = 80.16007988 x2 = 80.16007988 Simplifying x2 = 80.16007988 Take the square root of each side: x = {-8.953216175, 8.953216175}

Subproblem 2

x2 + -40.1 = -40.06007988 Simplifying x2 + -40.1 = -40.06007988 Reorder the terms: -40.1 + x2 = -40.06007988 Solving -40.1 + x2 = -40.06007988 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '40.1' to each side of the equation. -40.1 + 40.1 + x2 = -40.06007988 + 40.1 Combine like terms: -40.1 + 40.1 = 0.0 0.0 + x2 = -40.06007988 + 40.1 x2 = -40.06007988 + 40.1 Combine like terms: -40.06007988 + 40.1 = 0.03992012 x2 = 0.03992012 Simplifying x2 = 0.03992012 Take the square root of each side: x = {-0.1998002, 0.1998002}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-8.953216175, 8.953216175, -0.1998002, 0.1998002}

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