3x^4-27x^2+50=0

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Solution for 3x^4-27x^2+50=0 equation:


Simplifying
3x4 + -27x2 + 50 = 0

Reorder the terms:
50 + -27x2 + 3x4 = 0

Solving
50 + -27x2 + 3x4 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
16.66666667 + -9x2 + x4 = 0

Move the constant term to the right:

Add '-16.66666667' to each side of the equation.
16.66666667 + -9x2 + -16.66666667 + x4 = 0 + -16.66666667

Reorder the terms:
16.66666667 + -16.66666667 + -9x2 + x4 = 0 + -16.66666667

Combine like terms: 16.66666667 + -16.66666667 = 0.00000000
0.00000000 + -9x2 + x4 = 0 + -16.66666667
-9x2 + x4 = 0 + -16.66666667

Combine like terms: 0 + -16.66666667 = -16.66666667
-9x2 + x4 = -16.66666667

The x term is -9x2.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x2 + 20.25 + x4 = -16.66666667 + 20.25

Reorder the terms:
20.25 + -9x2 + x4 = -16.66666667 + 20.25

Combine like terms: -16.66666667 + 20.25 = 3.58333333
20.25 + -9x2 + x4 = 3.58333333

Factor a perfect square on the left side:
(x2 + -4.5)(x2 + -4.5) = 3.58333333

Calculate the square root of the right side: 1.892969448

Break this problem into two subproblems by setting 
(x2 + -4.5) equal to 1.892969448 and -1.892969448.

Subproblem 1

x2 + -4.5 = 1.892969448 Simplifying x2 + -4.5 = 1.892969448 Reorder the terms: -4.5 + x2 = 1.892969448 Solving -4.5 + x2 = 1.892969448 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x2 = 1.892969448 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = 1.892969448 + 4.5 x2 = 1.892969448 + 4.5 Combine like terms: 1.892969448 + 4.5 = 6.392969448 x2 = 6.392969448 Simplifying x2 = 6.392969448 Take the square root of each side: x = {-2.528432211, 2.528432211}

Subproblem 2

x2 + -4.5 = -1.892969448 Simplifying x2 + -4.5 = -1.892969448 Reorder the terms: -4.5 + x2 = -1.892969448 Solving -4.5 + x2 = -1.892969448 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x2 = -1.892969448 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x2 = -1.892969448 + 4.5 x2 = -1.892969448 + 4.5 Combine like terms: -1.892969448 + 4.5 = 2.607030552 x2 = 2.607030552 Simplifying x2 = 2.607030552 Take the square root of each side: x = {-1.61463016, 1.61463016}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.528432211, 2.528432211, -1.61463016, 1.61463016}

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