27x^4-208x^2+320=0

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Solution for 27x^4-208x^2+320=0 equation:


Simplifying
27x4 + -208x2 + 320 = 0

Reorder the terms:
320 + -208x2 + 27x4 = 0

Solving
320 + -208x2 + 27x4 = 0

Solving for variable 'x'.

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

Divide each side by '27'.
11.85185185 + -7.703703704x2 + x4 = 0

Move the constant term to the right:

Add '-11.85185185' to each side of the equation.
11.85185185 + -7.703703704x2 + -11.85185185 + x4 = 0 + -11.85185185

Reorder the terms:
11.85185185 + -11.85185185 + -7.703703704x2 + x4 = 0 + -11.85185185

Combine like terms: 11.85185185 + -11.85185185 = 0.00000000
0.00000000 + -7.703703704x2 + x4 = 0 + -11.85185185
-7.703703704x2 + x4 = 0 + -11.85185185

Combine like terms: 0 + -11.85185185 = -11.85185185
-7.703703704x2 + x4 = -11.85185185

The x term is -7.703703704x2.  Take half its coefficient (-3.851851852).
Square it (14.83676269) and add it to both sides.

Add '14.83676269' to each side of the equation.
-7.703703704x2 + 14.83676269 + x4 = -11.85185185 + 14.83676269

Reorder the terms:
14.83676269 + -7.703703704x2 + x4 = -11.85185185 + 14.83676269

Combine like terms: -11.85185185 + 14.83676269 = 2.98491084
14.83676269 + -7.703703704x2 + x4 = 2.98491084

Factor a perfect square on the left side:
(x2 + -3.851851852)(x2 + -3.851851852) = 2.98491084

Calculate the square root of the right side: 1.727689451

Break this problem into two subproblems by setting 
(x2 + -3.851851852) equal to 1.727689451 and -1.727689451.

Subproblem 1

x2 + -3.851851852 = 1.727689451 Simplifying x2 + -3.851851852 = 1.727689451 Reorder the terms: -3.851851852 + x2 = 1.727689451 Solving -3.851851852 + x2 = 1.727689451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.851851852' to each side of the equation. -3.851851852 + 3.851851852 + x2 = 1.727689451 + 3.851851852 Combine like terms: -3.851851852 + 3.851851852 = 0.000000000 0.000000000 + x2 = 1.727689451 + 3.851851852 x2 = 1.727689451 + 3.851851852 Combine like terms: 1.727689451 + 3.851851852 = 5.579541303 x2 = 5.579541303 Simplifying x2 = 5.579541303 Take the square root of each side: x = {-2.362105269, 2.362105269}

Subproblem 2

x2 + -3.851851852 = -1.727689451 Simplifying x2 + -3.851851852 = -1.727689451 Reorder the terms: -3.851851852 + x2 = -1.727689451 Solving -3.851851852 + x2 = -1.727689451 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.851851852' to each side of the equation. -3.851851852 + 3.851851852 + x2 = -1.727689451 + 3.851851852 Combine like terms: -3.851851852 + 3.851851852 = 0.000000000 0.000000000 + x2 = -1.727689451 + 3.851851852 x2 = -1.727689451 + 3.851851852 Combine like terms: -1.727689451 + 3.851851852 = 2.124162401 x2 = 2.124162401 Simplifying x2 = 2.124162401 Take the square root of each side: x = {-1.457450651, 1.457450651}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.362105269, 2.362105269, -1.457450651, 1.457450651}

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