20x^4-53x^2+20=0

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Solution for 20x^4-53x^2+20=0 equation:


Simplifying
20x4 + -53x2 + 20 = 0

Reorder the terms:
20 + -53x2 + 20x4 = 0

Solving
20 + -53x2 + 20x4 = 0

Solving for variable 'x'.

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

Divide each side by '20'.
1 + -2.65x2 + x4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.65x2 + -1 + x4 = 0 + -1

Reorder the terms:
1 + -1 + -2.65x2 + x4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.65x2 + x4 = 0 + -1
-2.65x2 + x4 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.65x2 + x4 = -1

The x term is -2.65x2.  Take half its coefficient (-1.325).
Square it (1.755625) and add it to both sides.

Add '1.755625' to each side of the equation.
-2.65x2 + 1.755625 + x4 = -1 + 1.755625

Reorder the terms:
1.755625 + -2.65x2 + x4 = -1 + 1.755625

Combine like terms: -1 + 1.755625 = 0.755625
1.755625 + -2.65x2 + x4 = 0.755625

Factor a perfect square on the left side:
(x2 + -1.325)(x2 + -1.325) = 0.755625

Calculate the square root of the right side: 0.869266933

Break this problem into two subproblems by setting 
(x2 + -1.325) equal to 0.869266933 and -0.869266933.

Subproblem 1

x2 + -1.325 = 0.869266933 Simplifying x2 + -1.325 = 0.869266933 Reorder the terms: -1.325 + x2 = 0.869266933 Solving -1.325 + x2 = 0.869266933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.325' to each side of the equation. -1.325 + 1.325 + x2 = 0.869266933 + 1.325 Combine like terms: -1.325 + 1.325 = 0.000 0.000 + x2 = 0.869266933 + 1.325 x2 = 0.869266933 + 1.325 Combine like terms: 0.869266933 + 1.325 = 2.194266933 x2 = 2.194266933 Simplifying x2 = 2.194266933 Take the square root of each side: x = {-1.48130582, 1.48130582}

Subproblem 2

x2 + -1.325 = -0.869266933 Simplifying x2 + -1.325 = -0.869266933 Reorder the terms: -1.325 + x2 = -0.869266933 Solving -1.325 + x2 = -0.869266933 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.325' to each side of the equation. -1.325 + 1.325 + x2 = -0.869266933 + 1.325 Combine like terms: -1.325 + 1.325 = 0.000 0.000 + x2 = -0.869266933 + 1.325 x2 = -0.869266933 + 1.325 Combine like terms: -0.869266933 + 1.325 = 0.455733067 x2 = 0.455733067 Simplifying x2 = 0.455733067 Take the square root of each side: x = {-0.675080045, 0.675080045}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.48130582, 1.48130582, -0.675080045, 0.675080045}

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